Uniform attractors for measure-driven quintic wave equations

In this paper we study the non-autonomous damped wave equation

$$\begin{equation} \begin{gathered} \, \partial_t^2 u+\gamma\,\partial_t u+(1-\Delta_x)u+f(u)=\mu(t),\quad t\geqslant\tau,\\ \{u,\partial_t u\}\big|_{t=\tau}=\{u_\tau,u_\tau'\}, \end{gathered} \end{equation} \tag{ 1.1 }$$

in the bounded subdomain $\mathbb{T}^3:=(-\pi,\pi)^3$ of $\mathbb{R}^3$ with periodic boundary conditions. Here $u(t,x)$ is the unknown function, $\Delta_x$ is the Laplacian with respect to the variable $x$, $\gamma$ is a positive constant, $f\colon \mathbb{R}\to\mathbb{R}$ is a given non-linearity which is assumed to be of quintic growth ($f(u)\sim u^5$ as $u\to\infty$) and to satisfy certain natural conditions (stated in (4.2) below), and $\mu$ is a given external force which is an $L^2$-valued measure of locally finite total variation and is assumed to be uniformly bounded on bounded time intervals: $\mu\in M_b(\mathbb{R},H)$ (see §2 for definitions of key function spaces).

Dispersive or/and dissipative semilinear wave equations of the form (1.1) model various oscillatory processes in many areas of modern mathematical physics including electrodynamics, quantum mechanics, non-linear elasticity, and so on, and constitute a topic of constant interest (see [26], [2], [40], [12], [36], [38], [35], and the references therein).

The basic property of these equations is the so-called energy equality

$$\begin{equation} \begin{gathered} \, \mathscr E(\xi_u(t))-\mathscr E(\xi_u(\tau))= -\gamma\int_\tau^t\|\partial_t u(s)\|^2_{L^2}\,ds+ \int_\tau^t(\partial_t u(s),\mu(ds)), \\ \nonumber \xi_u(t):=\{u(t),\partial_t u(t)\}, \end{gathered} \end{equation} \tag{ 1.2 }$$

which can formally be obtained by multiplying (1.1) by $\partial_t u$ and integrating with respect to $t$ and $x$. Here

$$\begin{equation*} \mathscr{E}(\xi_u):=\frac{1}{2}\bigl(\|\partial_t u\|^2_{L^2}+ \|\nabla_x u\|^2_{L^2}+\|u\|^2_{L^2}+2(F(u),1)\bigr),\qquad F(u):=\int_0^u f(z)\,dz, \end{equation*}$$

and $(u,v):=\int_{\mathbb{T}^3}u(x)v(x)\,dx$. This identity motivates the natural choice of the energy phase space and the class of energy solutions (as solutions for which the energy functional is finite) and also makes it possible to control the energy norm of the solution. In particular, if the non-linearity has a sub-quintic or quintic growth rate, then due to the Sobolev embedding theorem $H^1\subset L^6$, the energy space is given by $\mathscr{E}:=H^1(\mathbb{T}^3)\times L^2(\mathbb{T}^3)$. In the supercritical case where $f(u)\sim u|u|^q$ with $q>4$, we need to take $\mathscr{E}:=(H^1(\mathbb{T}^3)\cap L^{q+2}(\mathbb{T}^3))\times L^2(\mathbb{T}^3)$ in order to guarantee the finiteness of the energy integral.

It is believed that the analytic properties and the dynamics as $t\to\infty$ of solutions of the damped wave equations (1.1) depend strongly on the growth rate of the non-linearity $f(u)$ as $u\to\infty$. Indeed, in the most studied case of a cubic ($q=2$) or sub-cubic ($q<2$) growth rate the above energy equality norm is sufficient to get the well-posedness of the boundary-value problem in the energy space and also the dissipativity and additional regularity of solutions, as well as to develop the corresponding attractor theory in both the autonomous and non-autonomous cases (see [1], [2], [12], [16], [25], [26], [29], [40], [43], and the references therein).

The case of super-cubic but sub-quintic growth rate ($2<q<4$) is a bit more complicated since the well-posedness of energy solutions is still an open problem (at least in the case of bounded domains). However, this problem can be overcome by using solutions slightly more regular than energy solutions, for example, solutions for which the mixed $L^4(\tau,T;L^{12}(\mathbb{T}^3))$ space-time norm is finite for any $T>\tau$. These are the so-called Shatah–Struwe (or Strichartz) solutions, for which the corresponding initial boundary-value problem is known to be well-posed. The existence of such solutions is strongly based on the Strichartz estimates for the linear wave equation (see Theorem 2.1 below), which are now available not only for the whole space, but also for bounded domains with Dirichlet or Neumann boundary conditions (see [3], [6], [9], [10], [36], [37], [38]). Moreover, crucial for the attractor theory is the following energy-to-Strichartz estimate for such solutions:

$$\begin{equation} \|u\|_{L^4(t,t+1;L^{12})}\leqslant Q(\|\xi_u(t)\|_{\mathscr{E}})+ Q(\|\mu\|_{L^1(t,t+1;L^2)}),\qquad t\geqslant\tau, \end{equation} \tag{ 1.3 }$$

where $Q$ is a monotone increasing function which is independent of $t$ and the solution $u$. In the sub-quintic case this estimate is a straightforward consequence of the linear Strichartz estimate and perturbation arguments. In turn, (1.3) enables us to establish the dissipativity of $u$ in the Strichartz norm based on standard energy estimates. Since finiteness of this norm is enough for uniqueness, the control obtained gives well-posedness, dissipativity, and the existence of global/uniform attractors as in the classical cubic case; see [14], [22], and [18] for the cases of $\mathbb{R}^3$, $\mathbb{T}^3$, and a bounded domain endowed with Dirichlet boundary conditions, respectively (see also [32] for the case of a damped wave equation with fractional damping).

In contrast to this, very little is known about the solutions of (1.1) in the case of supercritical (superquintic) growth rate of the non-linearity $f$. In this case the situation is in some sense close to the 3D Navier–Stokes problem, namely, we have global existence of weak energy solutions, for which we do not know uniqueness, and local existence of more regular solutions, for which we do not know global existence. It is expected that smooth solutions may blow up in finite time even in the defocusing case, but to the best of our knowledge there are no such examples. In this case the existing attractor theory is related to multivalued semigroups or/and the so-called trajectory dynamical systems and trajectory attractors (see [12], [11], [29], [44], and the references therein).

We now turn to the most interesting borderline case of a critical quintic non- linearity $f$, which is the main object of our study in this paper. In this case, the energy-to-Strichartz estimate (1.3) no longer follows from the Strichartz estimate for the linear equation (at least in a straightforward way), so the proof of global existence for Shatah–Struwe solutions is usually based on so-called non-concentration arguments and the Pohozhaev–Morawetz equality; see [3], [15], [19]–[21], [34], [33], [35], and [38] (see also [9] and [10] for the case of bounded domains with Dirichlet or Neumann boundary conditions). This approach enables us to construct a Shatah–Struwe solution such that the $L^4(\tau,T;L^{12})$-norm is finite for all $T$, but does not let us get any control of the norm in terms of the energy norm or verify that the Strichartz norm does not grow as $T\to\infty$. This is clearly not sufficient for constructing and studying attractors. Indeed, without a uniform control of the Strichartz norm as $T\to\infty$, this extra regularity may a priori be lost in the limit, and the attractor may contain solutions which are less regular than Shatah–Struwe solutions (and for which we do not have a uniqueness theorem). Thus, a uniform control of the Strichartz norm is crucial for constructing an attractor theory.

This problem was partly overcome in [18], where the asymptotic regularity for autonomous damped quintic wave equations in bounded domains of $\mathbb{R}^3$ was established, along with the existence and regularity of global attractors. The method proposed there is heavily based on the existence of a global Lyapunov function and on the related convergence of the trajectories to the set of equilibria, and for this reason cannot be extended to the non-autonomous case. Moreover, to the best of our knowledge there have been no results so far on strong attractor theory for quintic wave equations in the non-autonomous case.

The main aim of our paper is to give a comprehensive study of non-autonomous quintic wave equations in the case of periodic boundary conditions. In order to do so, we first prove the energy-to-Strichartz estimate (1.3) for the Shatah–Struwe solutions of (1.1) for the quintic case as well. Therefore, the following theorem can be considered as our first main result.

Theorem 1.1.  Let the non-linearity $f$ satisfy the assumptions (4.2), and let the external force $\mu$ belong to $M_b(\mathbb{R},L^2)$. Then the problem (1.1) is globally well-posed in the class of Shatah–Struwe solutions, any such solution $u(t)$ satisfies the energy- to-Strichartz estimate (1.3), and the following dissipative estimate holds:

$$\begin{equation} \|\xi_u(t)\|_{\mathscr{E}}+\|u\|_{L^4(t,t+1;L^{12})}\leqslant Q(\|\xi_u(\tau)\|_{\mathscr{E}})e^{-\delta (t-\tau)}+ Q(\|\mu\|_{M_b(\mathbb{R},L^2)}), \end{equation} \tag{ 1.4 }$$

where the positive constant $\delta$ and the monotone increasing function $Q$ are independent of $t$, $\tau$, and the solution $u$.

The non-trivial part here is exactly to establish the energy-to-Strichartz estimate (1.3) (the rest is a standard consequence of this estimate and the classical dissipative energy estimate). To do so we start with the analogous energy-to-Strichartz estimate for the Shatah–Struwe solutions of the quintic wave equation in the whole space $\mathbb{R}^3$:

$$\begin{equation} \partial_t^2v-\Delta_x v+v^5=0, \end{equation} \tag{ 1.5 }$$

which was proved in [4] (see also [39] for the explicit expression of the function $Q$), and we extend it to the non-autonomous case

$$\begin{equation*} \partial_t^2 v-\Delta_x v+v^5=\mu(t). \end{equation*}$$

This extension uses an approximation of the external force $\mu(t)$ by sums of Dirac $\delta$-measures and a representation of the solution $v$ for such external forces via solutions of the autonomous equation. This approach can be interpreted as an analogue of the Duhamel formula for the non-linear equation and is of independent interest. We would like to emphasise that this method requires one to consider measure-driven equations of the form (1.1) as an intermediate step, even if we want to verify the estimate (1.4) only for regular external forces $\mu\in L^1_b(\mathbb{R},L^2)$ (see §5 for details). This is one of the motivations for us to consider measure-driven damped wave equations. Of course, measure-driven equations are interesting and important in themselves; we mention here only that they are widely used in the theory of stochastic partial differential equations (see [24] and the references therein). Note also that no analogue of the energy-to-Strichartz estimate for the case of equation (1.5) in bounded domains (with Dirichlet or Neumann boundary conditions) is known so far, and this is the main reason for our choice of periodic boundary conditions.

We now turn to attractor theory. We first note that the dissipative estimate (1.4) implies in a standard way the existence of a uniform attractor $\mathscr A_{\mathrm{un}}$ for equation (1.1) in the weak topology of the energy space $\mathscr{E}$ (see §6). However, new difficulties arise when we try to describe the uniform attractor in terms of bounded complete trajectories related to (1.1). We recall that, following the general theory developed in [12], [11], in order to obtain such a description we need to study not only equation (1.1), but also all its time shifts as well as their limits in the corresponding topology. In our case it is natural to take the closure of the set of all time shifts of the initial measure $\mu$ in the weak-star topology generated by the duality

$$\begin{equation*} M_{\rm loc}(\mathbb{R},L^2)=[C_{00}(\mathbb{R},L^2)]^*, \end{equation*}$$

where $C_{00}$ stands for the continuous functions with compact support. Namely, we introduce the group of time shifts $T(h)\colon M_b(\mathbb{R},L^2)\to M_b(\mathbb{R},L^2)$ via $(T(h)\mu)(t)=\mu(t+h)$, and we define the hull of the given measure $\mu$ as follows:

$$\begin{equation*} \mathscr H(\mu):= [\{T(h)\mu,\ h\in\mathbb{R}\}]_{M_{\rm loc}^{w^*}(\mathbb{R},L^2)} \end{equation*}$$

(see §6 for more details). Then the general theory predicts the representation

$$\begin{equation} \mathscr A_{\rm un}=\bigcup_{z\in\mathscr H(\mu)}\mathscr K_z\big|_{t=0}, \end{equation} \tag{ 1.6 }$$

where $\mathscr K_z$ is the set of complete (defined for all $t\in\mathbb{R}$) bounded (in $\mathscr{E}$) solutions of equation (1.1) with the right-hand side $z$. Again, according to the general theory, this representation will hold if the solution operators $U_z(t,\tau)$ (which map the initial data $\xi_\tau$ to the Shatah–Struwe solution $\xi_u(t)$ of the problem (1.1) with right-hand side $z\in\mathscr H(\mu)$) are weak-star continuous as maps from $\mathscr E\times\mathscr H(\mu)$ to $\mathscr{E}$.

Unfortunately, in contrast to the standard situations considered in [12], the map $z\to U_z(t,\tau)$ may be discontinuous in the case of measure-driven equations. As shown in §6, this may destroy (and destroys in concrete examples given there) the representation formula (1.6). More precisely, the attractor $\mathscr A_{\mathrm{un}}$ may become larger than the union of all bounded complete trajectories. In order to avoid this pathology, we found necessary and sufficient conditions for the measure $\mu$ which guarantee the continuity of the map $z\to U_z(t,\tau)$. In particular, because of these restrictions the measures $z\in \mathscr H(\mu)$ must have zero discrete part. For this reason we call these measures weakly uniformly non-atomic measures (see §6 for details). Thus, we have proved the following result.

Theorem 1.2.  Let the assumptions of Theorem 1.1 hold, and let the measure $\mu$ be weakly uniformly non-atomic. Then the weak uniform attractor $\mathscr A_{\mathrm{un}}$ has the representation (1.6).

We would like to recall that the representation formula (1.6) is one of the key tools for the further study of the attractor (and is crucial for our study of the compactness of weak attractors in stronger topologies; see §7). Unfortunately, this formula fails for generic measures $\mu\in M_b(\mathbb{R},L^2)$, which makes the theory constructed not entirely satisfactory. We expect that the problem can be resolved by using the trajectory approach, and we will return to this question in a forthcoming paper.

We would also like to mention that measure-driven equations appear naturally in attractor theory even if we start from a regular external force $\mu\in L^1_b(\mathbb{R},L^2)$ (the natural class of external forces from the point of view of Strichartz estimates). Indeed, we cannot guarantee in general that the hull $\mathscr H(\mu)$ is a subset of $L^1_b(\mathbb{R},L^2)$, and the appearance of Borel measures which are not absolutely continuous with respect to Lebesgue measure in the hull $\mathscr H(\mu)$ looks unavoidable. This is a second motivation for us to consider measure-driven damped wave equations from the very beginning.

As the next step, we study the existence of a uniform attractor for (1.1) in the strong topology of the energy space $\mathscr{E}$. Clearly, the assumption $\mu\in M_b(\mathbb{R},L^2)$ alone is not enough for this (see the examples given in [45]), so we need to impose some extra conditions on the measure $\mu$ to get this result. In this paper we introduce, following [45], two classes of right-hand sides, the so-called space-regular measures and the time-regular measures. Roughly speaking, these classes consist of the measures which can be approximated (in $M_b(\mathbb{R},L^2)$) by measures which are smooth in space or in time, respectively; see Definition 7.1. The intersection of these classes coincides with the class of translation-compact external forces introduced in [12].

The following result is verified in §7.

Theorem 1.3.  Let the assumptions of Theorem 1.2 hold, and in addition let the measure $\mu$ be space- or time-regular. Then there exists a uniform attractor for equation (1.1) in the strong topology of the energy space $\mathscr{E}$, and it coincides with the weak attractor $\mathscr A_{\mathrm{un}}$ constructed above.

As in [45], we utilise the energy equality and the so-called energy method (see also [5], [30]) to verify the asymptotic compactness.

Furthermore, we also verify that the uniform attractor is more smooth if the external forces are more smooth. As usual, in order to do so it is enough to verify that $\mathscr A_{\mathrm{un}}$ belongs to the higher energy space $\mathscr{E}^\alpha$ for some small positive $\alpha$. The additional regularity can be obtained using standard bootstrap arguments. To get this higher regularity, we follow mainly [43] and use the following consequence of the Kato–Ponce inequality:

$$\begin{equation*} \begin{aligned} \, \|f(v+w)-f(v)\|_{H^\alpha}&\leqslant C(1+\|v\|_{L^{12}}+ \|w\|_{L^{12}})^{4-\alpha} \\ &\qquad\times(1+\|v\|_{H^1}+\|w\|_{H^1})^{\alpha} \|w\|_{H^{1+\alpha}}^{1-\alpha}\|w\|_{H^{\alpha,12}}^{\alpha}, \end{aligned} \end{equation*}$$

which holds for $\alpha\in[0,2/5]$ (see §10). This enables us to prove the following result (in §8).

Theorem 1.4.  Let the assumptions of Theorem 1.1 hold, and in addition let

$$\begin{equation*} \mu\in M_b(\mathbb{R},H^\alpha) \end{equation*}$$

for some $\alpha\in(0,2/5]$. Then the attractor $\mathscr A_{\mathrm{un}}$ is a bounded set in the higher energy space $\mathscr{E}^\alpha$. Moreover, the analogous result holds also if $\mu$ is sufficiently smooth in time.

Finally, for the convenience of the reader we collect in §9 some standard facts and concepts in the theory of vector-valued measures and related functions of bounded variation.

In this section, we introduce some notation which will be used throughout the paper, and we state some classical results on solutions of linear wave equations. We start with function spaces.

Let $\Omega$ be a domain of $\mathbb{R}^3$ with a smooth boundary. As usual, the Lebesgue spaces of $p$-integrable functions on $\Omega$ are denoted by $L^p(\Omega)$, $1\leqslant p\leqslant\infty$. In the particular case $p=2$ we use the notation $H:=L^2(\Omega)$. For any $l\in\mathbb N$, we denote by $H^{l,p}(\Omega)=W^{l,p}(\Omega)$ the classical Sobolev space of distributions whose derivatives up to order $l$ belong to $L^p(\Omega)$. The closure of $C_0^\infty(\Omega)$ in the space $H^{l,p}(\Omega)$ is denoted by $H^{l,p}_0(\Omega)$. In the case $p=2$, we will write $H^l$ instead of $H^{l,2}$ in order to simplify the notation. The negative Sobolev spaces $H^{-l,p}(\Omega)$ are defined as dual spaces:

$$\begin{equation*} H^{-l,p}(\Omega)=\bigl(H^{l,q}_0(\Omega)\bigr)^*,\qquad \frac{1}{p}+\frac{1}{q}=1. \end{equation*}$$

In the case $l>0$, $l\notin\mathbb{N}$, we define the fractional space $H^{l,p}(\Omega)$, $1<p<\infty$, to be the restriction of the Bessel potentials space $H^{l,p}(\mathbb{R}^3)$ to the domain $\Omega$. We recall that the norm in $H^{l,p}(\mathbb{R}^3)$ is defined by

$$\begin{equation*} \|u\|_{H^{l,p}}:=\|(1+|\xi|^2)^{l/2}\widehat u(\xi)\|_{L^p}, \end{equation*}$$

where $\widehat u$ stands for the Fourier transform of $u$ (for instance, see [41] for more details). In particular, the fractional Laplacian gives an isomorphism between the spaces $H^{l,p}(\mathbb{R}^3)$ and $L^p(\mathbb{R}^3)$:

$$\begin{equation} (1-\Delta_x)^{l/2}H^{l,p}(\mathbb{R}^3)=L^p(\mathbb{R}^3). \end{equation} \tag{ 2.1 }$$

Note that this formula remains true in the spatially periodic case, when $\Omega=\mathbb{T}^3$. In the general case where $\Omega$ is a bounded domain some restrictions appear due to the boundary conditions (see [41]).

Below we will also make wide use of the classical Sobolev embedding theorem

$$\begin{equation*} H^{\alpha,p}(\mathbb{T}^3)\subset L^q(\mathbb{T}^3),\qquad \frac{1}{q}\geqslant \frac{1}{p}-\frac{\alpha}{3}\,,\quad \alpha\geqslant0,\quad q\ne\infty, \end{equation*}$$

and the interpolation inequality

$$\begin{equation*} \|u\|_{H^{\alpha,p}}\leqslant C\|u\|_{H^{\alpha_1,p_1}}^s\|u\|_{H^{\alpha_2,p_2}}^{1-s}, \end{equation*}$$

where $\alpha_1,\alpha_2\in\mathbb{R}$, $1<p_1,p_2<\infty$, $s\in[0,1]$, and

$$\begin{equation*} \alpha=s\alpha_1+(1-s)\alpha_2,\quad \frac{1}{p}=\frac{s}{p_1}+\frac{1-s}{p_2}\,. \end{equation*}$$

We will also need spaces of functions of mixed space-time regularity. For instance, the natural norms in the spaces $L^p(a,b;H^{\alpha,q})$ and $H^{1,p}(a,b; H^{\alpha,q})$ are given by

$$\begin{equation*} \|u\|_{L^p(a,b;H^{\alpha,q})}^p:= \int_a^b\|u(t,\,\cdot\,)\|_{H^{\alpha,q}}^p\,dt \end{equation*}$$

and

$$\begin{equation*} \|u\|_{H^{1,p}(a,b;H^{\alpha,q})}^p:= \int_a^b\bigl(\|u(t,\,\cdot\,)\|_{H^{\alpha,q}}^p+ \|\partial_t u(t,\,\cdot\,)\|_{H^{\alpha,q}}^p\bigr)\,dt, \end{equation*}$$

respectively. The index 'loc' or '$b$' stands for the local or uniformly local topology, respectively. For instance,

$$\begin{equation*} L^p_{\rm loc}(\mathbb{R}, H^{\alpha,q}):= \bigl\{u\colon\mathbb{R}\to H^{\alpha,q}\colon \|u\|_{L^p(a,b;H^{\alpha,q})}<\infty \ \forall\,[a,b]\subset\mathbb{R}\bigr\} \end{equation*}$$

and

$$\begin{equation*} L^p_b(\mathbb{R},H^{\alpha,q}):= \Bigl\{u\in L^p_{\rm loc}(\mathbb{R},H^{\alpha,q}): \|u\|_{L^p_b(\mathbb{R},H^{\alpha,q})}:= \sup_{a\in\mathbb{R}}\|u\|_{L^p_b(a,a+1;H^{\alpha,q})}<\infty\Bigr\}. \end{equation*}$$

Finally, to treat external forces, we need the space $M(a,b;H)$ of vector measures with values in $H$ and with finite total variation, and the associated spaces $\mathrm{BV}(a,b;H)$ of functions of bounded variation (see §9 for more details). Namely, $M_{\mathrm{loc}}(\mathbb{R},H)$ denotes the locally convex space of $H$-valued Borel measures $\nu$ on $\mathbb{R}$ such that the restrictions of $\nu$ to every finite segment $[s,t]$ belong to $M(s,t;H)$. Similarly,

$$\begin{equation*} M_b(\mathbb{R},H)=\Bigl\{\nu\in M_{\rm loc}(\mathbb{R},H)\colon \|\nu\|_{M_b(\mathbb{R},H)}:=\sup_{t\in\mathbb{R}}\|\nu\|_{M(t,t+1;H)}< \infty\Bigr\}. \end{equation*}$$

The spaces $\mathrm{BV}_{\mathrm{loc}}(\mathbb{R},H)$ and $\mathrm{BV}_b(\mathbb{R},H)$ are defined similarly.

We now recall standard results about the solutions of the linear wave equation

$$\begin{equation} \partial_t^2 v-\Delta_x v=g(t),\qquad \xi_v\big|_{t=0}=\xi_0, \end{equation} \tag{ 2.2 }$$

in the energy phase spaces

$$\begin{equation*} \xi_v(t):=\{v(t),\partial_t v(t)\}\in \mathscr{E}^\alpha:= H^{1+\alpha}(\mathbb{T}^3)\times H^\alpha(\mathbb{T}^3). \end{equation*}$$

For simplicity, we state the results for the spatially periodic case, although most of the results stated below remain true in the case of general bounded domains as well.

Theorem 2.1.  Let the initial data satisfy $\xi_0\in\mathscr E^\alpha$, let $T>0$, and let $g(t)\in L^1([0,T];H^\alpha)$ for some $\alpha\in\mathbb{R}$. Then there is a unique solution $\xi_v\in C(0,T;\mathscr{E}^\alpha)$ of the problem (2.2). Also, the solution $v$ belongs to the space $L^4(0,T;H^{\alpha,12}(\mathbb{T}^3))$ and the following estimate holds:

$$\begin{equation} \|\xi_v\|_{C(0,T;\mathscr{E}^\alpha)}+ \|v\|_{L^4(0,T;H^{\alpha,12}(\mathbb T^3))}\leqslant C_T(\|\xi_0\|_{\mathscr{E}^\alpha}+\|g\|_{L^1(0,T;H^\alpha)}), \end{equation} \tag{ 2.3 }$$

where the constant $C_T$ does not depend on $\xi_0\in\mathscr{E}^\alpha$ and $g(t)\in L^1([0,T];H^\alpha)$.

The proof of this theorem can be found in [3], [35], or [38], for example.

To conclude this section, we state the analogue of the above estimate for the damped linear wave equation

$$\begin{equation} \partial_t^2 v+\gamma\,\partial_t v+(1-\Delta_x)v=g(t),\quad \xi_v\big|_{t=\tau}=\xi_\tau, \end{equation} \tag{ 2.4 }$$

where $\gamma>0$. This estimate will be crucially used later on in order to obtain an additional regularity of uniform attractors.

Corollary 2.2.  Let $\xi_\tau\in \mathscr{E}^\alpha$ and $g\in L^1_{\mathrm{loc}}(\mathbb{R},H^\alpha)$ for some $\alpha\in\mathbb{R}$. Then the solution $\xi_v(t)$ of (2.4) satisfies the estimate

$$\begin{equation} \begin{aligned} \, \nonumber &\|\xi_v(t)\|_{\mathscr{E}^\alpha}+\biggl(\int_\tau^t e^{-4\delta(t-s)} \|v(s)\|^4_{H^{\alpha,12}}\,ds\biggr)^{1/4} \\ &\qquad\leqslant C\|\xi_\tau\|_{\mathscr{E}^{\alpha}}e^{-\delta(t-\tau)}+ C\int_\tau^te^{-\delta(t-s)}\|g(s)\|_{H^\alpha}\,ds, \end{aligned} \end{equation} \tag{ 2.5 }$$

where the positive constants $C$ and $\delta=\delta(\gamma)$ are independent of $t\geqslant\tau$, $\xi_\tau$, and $g$.

Proof.  Indeed, due to the isomorphism (2.1) it is sufficient to verify (2.5) just for $\alpha=0$. For simplicity we also assume that $\tau=0$. Multiplying (2.4) by $\partial_t v+\beta v$, where $\beta>0$ is sufficiently small, and arguing in the standard way (see [12], for example), we arrive at the inequality

$$\begin{equation} \|\xi_v(t)\|_{\mathscr{E}}\leqslant C\|\xi_\tau\|_{\mathscr{E}}e^{-\delta t}+ C\int_0^te^{-\delta(t-s)}\|g(s)\|_H\,ds \end{equation} \tag{ 2.6 }$$

for some positive constants $C$ and $\delta$. After that, we rewrite (2.4) in the form of the equation (2.2) with the right-hand side

$$\begin{equation*} \widetilde g(t)=g(t)-\gamma\partial_t v(t)-v(t) \end{equation*}$$

and apply the estimate (2.3) for the Strichartz norm on the time interval $[t,t+1]$, $t\geqslant0$. This gives us the estimate

$$\begin{equation} \begin{aligned} \, \nonumber \|v\|_{L^4(t,t+1;L^{12})}&\leqslant C\Bigl(\,\sup_{s\in[t,t+1]} \|\xi_v(s)\|_{\mathscr E}+\|g\|_{L^1(t,t+1;H)}\Bigr) \\ &\leqslant C\|\xi_0\|_{\mathscr{E}}e^{-\delta t}+ C\int_0^{t+1}e^{-\delta(t-s)}\|g(s)\|_H\,ds. \end{aligned} \end{equation} \tag{ 2.7 }$$

We claim that (2.7) implies the estimate (2.5) for the Strichartz norm. Indeed, we may assume without loss of generality that $t=n\in\mathbb{N}$ (if this condition is not satisfied, then we can always increase $t$ by a suitable $\kappa\in(0,1)$ to satisfy this assumption and put $g(s)=0$ for $s\geqslant t$). In this case, using the concavity of the function $z^{1/4}$ and (2.7), we get that for some $0<\delta'<\delta$

$$\begin{equation} \begin{aligned} \, \nonumber &\biggl(\,\int_0^te^{-4\delta'(t-s)}\|v\|^4_{L^{12}}\,ds\biggr)^{1/4}\leqslant C\biggl(\,\sum_{k=0}^{n-1}e^{-4\delta'(n-k)}\int_{k}^{k+1} \|v(s)\|_{L^{12}}^4\,ds\biggr)^{1/4} \\ \nonumber &\qquad\leqslant C\sum_{k=0}^{n-1}e^{-\delta'(n-k)} \|v\|_{L^4(k,k+1;L^{12})} \\ \nonumber &\qquad\leqslant C\sum_{k=0}^{n-1}e^{-\delta'(n-k)}e^{-\delta k} \|\xi_v(0)\|_{\mathscr{E}}+C\sum_{k=0}^{n-1}e^{-\delta'(n-k)} \int_0^{k+1}e^{-\delta(k+1-s)}\|g(s)\|_H\,ds \\ \nonumber &\qquad\leqslant Ce^{-\delta't}\|\xi_v(0)\|_{\mathscr{E}} +C\int_0^te^{-\delta'(t-s)}\sum_{k=1}^{n-1}e^{-(\delta-\delta')|k-s|} \|g(s)\|_H\,ds \\ &\qquad\leqslant C\|\xi_v(0)\|_{\mathscr{E}} e^{-\delta't} +C(\delta-\delta')^{-1}\int_0^te^{-\delta'(t-s)}\|g(s)\|_H\,ds. \end{aligned} \end{equation} \tag{ 2.8 }$$

Finally, replacing $\delta$ by $\delta'$, we get the desired estimate for the Strichartz norm and finish the proof of the corollary. $\square$

In this section we consider the linear wave equation

$$\begin{equation} \partial_t^2 w+\gamma\partial_t w+(-\Delta_x+1)w=\mu(t),\qquad \xi_w\big|_{t=\tau}=\xi_\tau:=\{w_\tau,w'_\tau\}, \end{equation} \tag{ 3.1 }$$

on a three-dimensional torus $x\in\mathbb{T}^3$, where the damping parameter $\gamma$ is $\geqslant 0$ and, in contrast to the previous section, $\mu$ is a measure. All the results in this section are actually valid not only for the case of periodic boundary conditions but also for Dirichlet or Neumann boundary conditions when $\Omega\subset \mathbb{R}^3$ is a smooth bounded domain (although this result is not necessary for our purposes). We suppose here that

$$\begin{equation} \mu \in M(\tau,T;H), \end{equation} \tag{ 3.2 }$$

where $M(\tau,T;H)$ is the space of $H$-valued Borel vector measures on $[\tau,T]$ with finite total variation (see §9 for more details).

We start with the definition of an energy solution of (3.1) which is a bit more delicate since, in contrast to the usual case, the time derivative $\partial_t w(t)$ may have jumps produced by atoms of the measure $\mu$.

Definition 3.1.  A function $w(t)$ such that $\xi_w(t)\in L^\infty(\tau,T;\mathscr{E})$ (where $\xi_w(t):=\{w(t),\partial_t w(t)\}$) is an energy solution of the problem (3.1) on $[\tau,T]$ if:

1) it satisfies the equation in the sense of distributions, that is, for any test function $\phi\in C^\infty_0((\tau,T)\times \mathbb{T}^3)$

$$\begin{equation} \begin{aligned} \, \nonumber &-\int_\tau^T(\partial_t w(t),\partial_t \phi(t))\,dt+ \int_\tau^T(\nabla w(t),\nabla \phi(t))\,dt+\int_\tau^T(w(t),\phi(t))\,dt \\ &\qquad+\gamma\int_\tau^T(\partial_t w(t),\phi(t))\,dt= \int_\tau^T(\phi(t),\mu(dt)); \end{aligned} \end{equation} \tag{ 3.3 }$$

2) $\xi_w(t)$ is weakly left-continuous at every point $t\in[\tau,T]$ as an $\mathscr E$-valued function;

3) the initial conditions are satisfied in the sense that

$$\begin{equation*} \xi_w(\tau)=\{w_\tau,w'_\tau\}\quad\text{and}\quad \xi_w(\tau+0):=\operatorname{w\text{-}\lim}_{s\to\tau+0}\xi_w(s)= \bigl\{w_\tau,w'_\tau+\mu(\{\tau\})\bigr\}. \end{equation*}$$

Remark 3.2.  Since $w\in L^\infty(\tau,T;H^1)$ and $\partial_t w\in L^\infty(\tau,T;H)$, the function $w(t)$ is weakly continuous as a function with values in $H^1$, $w\in C_\mathrm{w}(\tau,T;H^1)$, so the initial data for $w(t)$ is well defined. The situation with the derivative $\partial_t w$ is a bit more delicate since it can be discontinuous. Namely, from Definition 3.1 we see that the distributional derivative $\partial_t^2w$ satisfies

$$\begin{equation} \begin{aligned} \, \nonumber \langle\partial_t^2 w,\phi\rangle&=-\langle\partial_t w,\partial_t\phi\rangle= -\int_\tau^T(\partial_t w(t),\partial_t \phi(t))\,dt \\ \nonumber &=-\int_\tau^T(\nabla w(t),\nabla \phi(t))\,dt-\int_\tau^T(w(t),\phi(t))\,dt \\ &\qquad-\gamma\int_\tau^T(\partial_t w(t),\phi(t))\,dt+ \int_\tau^T(\phi(t),\mu(dt)), \end{aligned} \end{equation} \tag{ 3.4 }$$

and this functional can clearly be extended by continuity to any $\phi\in L^1(\tau,T;H^1)\cap C_0(\tau,T;H)$. For this reason,

$$\begin{equation*} \partial_t^2 w\in L^\infty(\tau,T;H^{-1})+M(\tau,T;H). \end{equation*}$$

This, together with the fact that $\partial_t w\in L^\infty(\tau,T;H)$, implies that

$$\begin{equation*} \partial_t w\in C_\mathrm{w}(\tau,T;H)+\mathrm{BV}(\tau,T;H). \end{equation*}$$

Since any $\mathrm{BV}$-function has left and right limits at every point (see §9), the function $t\mapsto\partial_t w(t)$ also has left and right limits $\partial_t w(t+0)$ and $\partial_t w(t-0)$ at any point $t\in(\tau,T)$ (in the weak topology of $H:=L^2$) as well as the limits $\partial_t w(\tau+0)$ and $\partial_t w(T-0)$. Thus, assumption 2) of the definition makes sense, and the second part of the initial conditions 3) for $\partial_t w$ is also well-defined. However, since $C_0(\tau,T;H)$ is not dense in $C(\tau,T;H)$, the values $\partial_t w(\tau)$ and $\partial_t w(T)$ remain undefined (as well as the values of $\partial_t w$ at the jump points).

In order to avoid this ambiguity and to be able to define the dynamical process associated with our problem (see §6), we choose a weakly left-continuous representative on $[\tau,T]$ from the equivalence class of $\partial_t w$ by default. Then the value $\partial_t w(T)=\partial_t w(T-0)$ is also well defined, and the value $\partial_t w(\tau)$ is determined by the first part of the initial conditions 3).

Remark 3.3.  Note that any energy solution $w$ of the problem (3.1) has the following property:

$$\begin{equation} \partial_t w(t+0)-\partial_t w(t)=\mu(\{t\}),\qquad t\in[\tau,T] \end{equation} \tag{ 3.5 }$$

(in the case $t=T$ we just assume that $\xi_w(T+0):=\{w(T),\partial_t w(T)+\mu(\{T\})\}$). Indeed, integrating by parts in (3.3) and using (9.22) to handle the most complicated term, which involves measures, we get that

$$\begin{equation} \int_\tau^T\bigl(W(t),\partial_t \phi(t)\bigr)\,dt=0\quad \forall\, \phi\in C^\infty_0((\tau,T)\times\mathbb{T}^3), \end{equation} \tag{ 3.6 }$$

where $W(t):=-\partial_t w(t)-\int_\tau^t(-\Delta_x+1)w(s)\,ds- \gamma w(t)+\mu([\tau,t))$. Therefore, $W(t)=\Psi$ almost everywhere for some $\Psi\in H^{-1}$. Using now the assumption that $\partial_t w$ is left-continuous, together with the obvious fact that $t\mapsto\mu([\tau,t))$ is also left-continuous, and taking into account the initial data, we conclude that

$$\begin{equation} \partial_t w(t)=-\int_\tau^t(-\Delta_x+1)w(s)\,ds-\gamma w(t)+\mu([\tau,t))+ w_\tau'+\gamma w_\tau,\quad t\in[\tau,T]. \end{equation} \tag{ 3.7 }$$

The desired formula (3.5) is an immediate consequence of (3.7).

The formula proved shows, in particular, that the function $\partial_t w$ will be weakly continuous as a function with values in $H$ if the measure $\mu$ is non-atomic. Moreover, multiplying (3.7) by $\partial_t\phi$, integrating with respect to $t\in\mathbb{R}$, integrating back by parts, and using the initial conditions, we return to the distributional formulation (3.3). Thus, the identities (3.3) and (3.7) are equivalent and we may verify (3.7) instead of (3.3). We will make essential use of this observation later.

In the next step we write out an explicit formula for the solution of (3.1). We start with the homogeneous case $\mu=0$. Then the solution $w(t)$ is given by

$$\begin{equation} w(t)=C_{A}(t-\tau)w_\tau+S_{A}(t-\tau)w'_\tau, \end{equation} \tag{ 3.8 }$$

where $A:=-\Delta_x+1$ with periodic boundary conditions,

$$\begin{equation*} S_A(t):=e^{-\gamma t/2}\,\frac{\sin(\Lambda(A) t)}{\Lambda(A)}\,,\qquad C_A(t):=e^{-\gamma t/2}\cos(\Lambda(A)t), \end{equation*}$$

and $\Lambda(z):=(z-\gamma^2/4)^{1/2}$. The corresponding solution semigroup in the energy phase space $\mathscr E$ is then defined by

$$\begin{equation} \xi_w(t)=\mathscr S_A(t)\xi_w(0),\qquad \mathscr S_A(t):=\begin{pmatrix} C_A(t) & S_A(t) \\ \partial_t C_A(t) & \partial_t S_A(t) \end{pmatrix}. \end{equation} \tag{ 3.9 }$$

The following result is well known and can be verified by straightforward calculations.

Lemma 3.4.  The operators $\mathscr S_A(t)$ are bounded in $\mathscr E$ and satisfy the estimate

$$\begin{equation} \|\mathscr S_A(t)\|_{\mathscr L(\mathscr E,\mathscr E)}\leqslant Ce^{-\gamma_0t}, \end{equation} \tag{ 3.10 }$$

where the non-negative constants $C$ and $\gamma_0$ can depend on $\gamma$, and $\gamma_0>0$ if $\gamma>0$.

Furthermore, in the regular case where the measure $\mu$ is absolutely continuous ($\mu(dt)=g(t)\,dt$ for some $g\in L^1(\tau,T;H)$), the solution of the non-homogeneous equation is given by the Duhamel formula:

$$\begin{equation*} \xi_w(t)=\mathscr S_A(t)\xi_w(\tau)+\int_\tau^t\mathscr S_A(t-s) \begin{pmatrix}0\\g(s)\end{pmatrix}\,ds. \end{equation*}$$

The next theorem shows that the analogue of this formula holds in the general case as well.

Theorem 3.5.  Let $\gamma\geqslant 0$ and $\xi_\tau\in\mathscr{E}$, and let the external force $\mu$ belong to $M(\tau,T;H)$. Then the problem (3.1) has a unique energy solution $w$ on $[\tau,T]$. This solution is given by

$$\begin{equation} \xi_w(t)=\mathscr S_A(t)\xi_\tau+\int_{s\in[\tau,t)}\mathscr S_{A}(t-s) \begin{pmatrix}0 \\ \rho_{\mu}(s)\end{pmatrix}|\mu|(ds),\qquad t\in [\tau,T], \end{equation} \tag{ 3.11 }$$

where $\rho_\mu \in L^1_{|\mu|}(\tau,T;H)$ denotes the density of $\mu$ with respect to the measure $|\mu|$ (see (9.17)).

Furthermore, the following energy estimate holds:

$$\begin{equation} \|\xi_w(t)\|_{\mathscr{E}}\leqslant C\biggl(\|\xi_\tau\|_{\mathscr{E}} e^{-\gamma_0(t-\tau)}+\int_\tau^te^{-\gamma_0(t-s)}\,|\mu|(ds)\biggr),\qquad t\in[\tau,T], \end{equation} \tag{ 3.12 }$$

for some constant $C$ depending only on $\gamma$.

Proof.  We note first that due to Lemma 3.4 the function $\xi_w(t)$ is well defined and belongs to the space $L^\infty(\tau,t;\mathscr{E})$ and satisfies the energy inequality (3.12) (here we have implicitly used that $\|\rho_\mu(t)\|_H=1$). The weak left-continuity of $\xi_w(t)$ as well as the fact that it satisfies the initial data is also an immediate consequence of (3.11).

In order to check that it satisfies the equation in the sense of distributions, we expand $\xi_w(t)$ into a Fourier series associated with the eigenfunctions of the operator $A$. Namely, let $e_i$ and $\lambda_i$ be the eigenvectors and eigenvalues (enumerated in non-decreasing order) of $A$, respectively, and let $P_N$ be the orthogonal projection onto the linear subspace spanned by the first $N$ eigenvectors. Also, let $Q_N:=1-P_N$. Then

$$\begin{equation*} \xi_w(t)=\xi_{P_Nw}(t)+\xi_{Q_N w}(t) \end{equation*}$$

and, by Lemma 9.6 and the estimate (3.12),

$$\begin{equation} \lim_{N\to\infty}\|\xi_{Q_Nw}\|_{L^\infty(\tau,T;H)}=0. \end{equation} \tag{ 3.13 }$$

Thus, it is enough to verify that for every $N\in\mathbb N$ the function $w_N(t):=P_Nw(t)$ is a distributional solution of the ordinary differential equation

$$\begin{equation*} \partial_t^2w_N+\gamma\,\partial_t w_N+Aw_N=P_N\mu. \end{equation*}$$

But this can be done in a straightforward way by using the integration by parts formula (9.22) (with $H=\mathbb{R}^N$) and the properties of the Duhamel integral (we leave the rigorous proof of this to the reader). Thus, the function $\xi_w(t)$ is indeed the desired energy solution.

Finally, let $w_1(t)$ and $w_2(t)$ be two energy solutions. Then since both these functions are weakly continuous in $H^1$, and their derivatives $\partial_t w_i(t)$ are weakly left-continuous and have the same jumps according to (3.5), we conclude that $\xi_w(t)$ is weakly continuous in $\mathscr E$, where $w(t)=w_1(t)-w_2(t)$. In addition, $w(t)$ solves the homogeneous problem (3.1) with $\mu=0$ and zero initial data. It is well known that such a solution is unique, so $w\equiv0$ and uniqueness is also verified. $\square$

Corollary 3.6.  Let the assumptions of Theorem 3.5 hold. Then the energy solution $w$ is in $C(\tau,T;H^1)$ and $\partial_t w\in C([\tau,T]\setminus \operatorname{supp}\mu_{\mathrm{d}};H)$, where $\operatorname{supp}\mu_{\mathrm{d}}$ is the support of the discrete part of the measure $\mu$, or equivalently, the set of points of discontinuity of the distribution function $\Phi_\mu(t)$. Moreover, both the limits $\xi_w(t+0)$ and $\xi_w(t-0)$ exist for every $t\in[\tau,T]$ in the strong topology of $\mathscr E$.

Indeed, this follows immediately from the analogous statement for the finite- dimensional part $\xi_{P_Nw}(t)$ and from the uniform smallness of the function $\xi_{Q_Nw}(t)$ proved in the theorem.

Corollary 3.7.  Assume that, in addition, the measure $\mu$ is non-atomic ($\mu(\{t\})=0$ for all $t$). Then the solution $\xi_w$ is in $C(\tau,T;\mathscr E)$. Moreover, the energy equality holds:

$$\begin{equation} \frac{1}{2}(\|\xi_w(t)\|_{\mathscr E}^2-\|\xi_w(s)\|_{\mathscr E}^2)= -\gamma\int_s^t\|\partial_t w(\kappa)\|^2_H\,d\kappa+ \int_s^t(\partial_t w(\kappa),\mu(d\kappa)) \end{equation} \tag{ 3.14 }$$

for all $[s,t]\subset[\tau,T]$ (here and below, $\int_\tau^t$ is written instead of $\int_{\kappa\in[\tau,t)}$, since the integrals over $[\tau,t)$ and $[\tau,t]$ coincide for all non-atomic measures).

As usual, (3.14) is proved first for the finite-dimensional function $\xi_{w_N}(t)$, where it is standard since $\xi_{w_N}(t)$ is continuous in time and therefore can be approximated by smooth functions. Then by passing to the limit as $N\to\infty$ one gets the desired energy equality also for the infinite-dimensional case (using the fact that the $\xi_{Q_Nw}$ are uniformly small).

Remark 3.8.  The identity (3.14) can be rewritten in the form

$$\begin{equation} \frac{d}{dt}\biggl(\frac{1}{2}\|\xi_w(t)\|^2_{\mathscr E}- \int_\tau^t(\partial_t w(s),\mu(ds))\biggr)= -\gamma\|\partial_t w(t)\|^2_H. \end{equation} \tag{ 3.15 }$$

In particular, the function $\dfrac{1}{2}\|\xi_w(t)\|^2_{\mathscr E}- \int_\tau^t(\partial_t w(s),\mu(ds))$ is absolutely continuous in time. However, the energy $\|\xi_w(t)\|^2_{\mathscr E}$ itself is not necessarily absolutely continuous since the singular part of the measure $\mu$ is not assumed to vanish.

The analogue of this formula can be written in the general case, when the discrete part of $\mu$ does not vanish. However, in this case one should be careful with the definition of the integral $\int_{[\tau,t)}(\partial_t w(\kappa),\mu_{\rm d}(d\kappa))$, since the function $\partial_t w$ makes jumps exactly at the points where $\Phi_\mu(t)$ is discontinuous. Moreover, since by (3.7) the function $\partial_t w-\mu([\tau,t))$ is continuous, the only problematic term is $\int_{[\tau,t)}(\mu_{\rm d}([\tau,s)),\mu_{\rm d}(ds))$. This integral makes sense as a Lebesgue–Stiltjes integral. But the value of the integral thus defined is inconsistent with the energy identity. Indeed, in our case $\partial_t w$ is left-continuous at the jump points $t=t_j$ and therefore

$$\begin{equation*} \int_{[\tau,t)}\bigl(\partial_t w(s),\mu_{\rm d}(ds)\bigr)= \sum_{j}\bigl(\partial_t w(t_j-0),\mu(\{t_j\})\bigr)_H. \end{equation*}$$

However, arguing a bit more carefully (for example, approximating $\mu_{\mathrm{d}}$ by smooth functions or comparing the values of the energy functional before and after a jump), we see that the correct formula must be

$$\begin{equation*} \int_{[\tau,t)}(\partial_t w(t),\mu_{\rm d}(ds)):= \sum_{j}\biggl(\frac{1}{2}(\partial_t w(t_j+0)+ \partial_t w(t_j-0)),\mu(\{t_j\})\biggr)_H, \end{equation*}$$

which corresponds to the choice $\partial_t w(t_j):= (\partial_t w(t_j+0)+\partial_t w(t_j-0))/2$ (see also [17]). This gives the following natural interpretation of the problematic integral:

$$\begin{equation*} \int_{[\tau,t)}(\partial_t w(s),\mu(ds)):= \int_{[\tau,t)}(\partial_t w(s)-\mu([\tau,s)),\mu(ds))+ \frac{1}{2}\|\mu([\tau,t))\|_H^2, \end{equation*}$$

which is consistent with the energy equality. We will return to this topic in a forthcoming paper.

We end this section by establishing Strichartz-type estimates for the measure- driven wave equation using approximations of the measure by absolutely continuous measures (functions in $L^1(\tau,T;H)$).

Theorem 3.9.  Let $\gamma\geqslant0$, let the initial data $\xi_\tau$ be in $\mathscr{E}$, and let the external force $\mu$ be in $M(\tau,T;H)$. Then the energy solution $w$ of the problem (3.1) has the estimate

$$\begin{equation} \|w\|_{L^4(\tau,T;L^{12})}\leqslant C(\|\xi_\tau\|_{\mathscr{E}}+\|\mu\|_{M(\tau,T;H)}), \end{equation} \tag{ 3.16 }$$

where the constant $C$ depends on $\gamma$ and $T-\tau$ but not on $\xi_\tau$ and $\mu$.

Proof.  Let $\Phi_\mu(t)$ be the distribution function of $\mu$ (see (9.5)). Let $\Phi_n(t)$ be smooth approximations of $\Phi_\mu$ constructed as in Proposition 9.11 and consider the following sequence $w_n$ of approximations of $w$:

$$\begin{equation} \begin{cases} \partial_t^2 w_n+\gamma\,\partial_t w-\Delta_x w_n+\gamma\,\partial_t w_n= \Phi'_n(t),& t\in[\tau,T], \\ \xi_{w_n}\big|_{t=\tau}=\xi_\tau.& \end{cases} \end{equation} \tag{ 3.17 }$$

We note that by construction (see Proposition 9.11) we have

$$\begin{equation} \mu_{\Phi_n}\to \mu\quad\text{as } n \to\infty\quad\text{weak-star in $M(\tau,T;H)$}, \end{equation} \tag{ 3.18 }$$

$$\begin{equation} \|\Phi'_n\|_{L^1(\tau,T;H)}\leqslant \|\mu\|_{M(\tau,T;H)}. \end{equation} \tag{ 3.19 }$$

In addition, $\mu_{\Phi_n}([\tau,t))\to\mu([\tau,t))$ for all $t\in[\tau,T]$ (see Remark 9.12). Using the standard energy estimate and (3.19), we see that

$$\begin{equation} \begin{aligned} \, \nonumber \|\xi_{w_n}\|_{L^\infty(\tau,T;\mathscr{E})}&\leqslant C(\|\xi_\tau\|_{\mathscr{E}}+\|\Phi'_n\|_{L^1(\tau,T;H)}) \\ &\leqslant C(\|\xi_\tau\|_{\mathscr{E}}+\|\mu\|_{M(\tau,T;H)})\quad \forall\,n\in\mathbb{N}. \end{aligned} \end{equation} \tag{ 3.20 }$$

The last estimate together with (3.18) implies that $\xi_{w_n}$ converges to some $\xi_{\overline w}\in L^\infty(\tau,T;\mathscr{E})$ weak-star in $L^\infty(\tau,T;\mathscr{E})$ as $n\to\infty$. We need to show that $\overline w$ is an energy solution of the problem (3.1). Indeed, arguing in the standard way we see that $w_n\to \overline w$ strongly in $C(\tau,T;H)$, and therefore $\overline w$ is weakly continuous in $H^1$ and $\overline w(\tau)=w_\tau$.

To verify that $\overline w$ is an energy solution, it is enough to take the pointwise limit in the equality

$$\begin{equation} \begin{aligned} \, \nonumber \partial_t w_n(t)&=-\int_\tau^t(-\Delta_x+1)w_n(s)\,ds \\ &\qquad-\gamma w_n(t)+\mu_{\Phi_n}([\tau,t))+w_\tau'+ \gamma w_\tau,\qquad t\in[\tau,T], \end{aligned} \end{equation} \tag{ 3.21 }$$

and get (3.7). Thus, $\overline w$ is an energy solution of (3.1), and by uniqueness $\overline w=w$.

To obtain the desired Strichartz estimate, we apply Theorem 2.1 to equation (3.17) and get that

$$\begin{equation} \begin{aligned} \, \nonumber \|w_n\|_{L^4(\tau,T;L^{12})}&\leqslant C(\|\xi_\tau\|_{\mathscr{E}}+ \|\Phi'_n\|_{L^1(\tau,T;H)}) \\ &\leqslant C(\|\xi_\tau\|_{\mathscr{E}}+\|\mu\|_{M(\tau,T;H)})\quad \forall\,n\in\mathbb{N}. \end{aligned} \end{equation} \tag{ 3.22 }$$

The last estimate allows us to assume without loss of generality that $w_n$ converges to $w$ weakly in $L^4(\tau,T;L^{12})$ as $n\to\infty$. Weak lower semicontinuity of the norm implies the desired estimate (3.16) and finishes the proof of the theorem. $\square$

Remark 3.10.  Since the energy estimate lets us control the $L^\infty(\tau,T;L^6)$-norm of $w$, we can replace the $L^4(\tau,T;L^{12})$-norm on the left-hand side of (3.16) by any intermediate Strichartz norm, for instance, by the $L^5(\tau,T;L^{10})$-norm.

In this section, we discuss the properties of solutions for our main object of study — the damped quintic wave equation

$$\begin{equation} \partial_t^2u+\gamma\,\partial_tu+(1-\Delta_x)u+f(u)=\mu,\quad \xi_u\big|_{t=\tau}=\xi_\tau \end{equation} \tag{ 4.1 }$$

on the 3D torus $\Omega=\mathbb T^3$. Since the results presented below are either well known or straightforward adaptations of well-known results to the case of external forces generated by measures, we restrict ourselves to giving only a brief exposition (more details can be found in [12], [18], [9], and [10]).

We assume that $\xi_\tau\in\mathscr E$, $\mu\in M_b(\mathbb{R},H)$, and the non-linearity $f\in C^2(\mathbb{R})$ has the following structure:

$$\begin{equation} f(u)=u^5+h(u),\qquad |h''(u)|\leqslant C(1+|u|^{q}),\quad q<3,\quad h(0)=0. \end{equation} \tag{ 4.2 }$$

We start our exposition by giving the analogue of Definition 3.1 for an energy solution in the non-linear case.

Definition 4.1.  A function $u(t)$ such that $\xi_u(t)\in L^\infty(\tau,T;\mathscr{E})$ (where $\xi_u(t):=\{u(t),\partial_t u(t)\}$) is an energy solution of the problem (4.1) on an interval $[\tau,T]$ if:

1) it satisfies equation (4.1) in the sense of distributions, that is, for any test function $\phi\in C^\infty_0((\tau,T)\times \Omega)$,

$$\begin{equation} \begin{aligned} \, \nonumber &-\int_\tau^T(\partial_t u(t),\partial_t \phi(t))\,dt+ \int_\tau^T(\nabla u(t),\nabla\phi(t))\,dt+\int_\tau^T(u(t),\phi(t))\,dt \\ &\qquad+\gamma\int_\tau^T(\partial_t u(t),\phi(t))\,dt+ \int_\tau^T(f(u(t)),\phi(t))\,dt=\int_\tau^T(\phi(t),\mu(dt)); \end{aligned} \end{equation} \tag{ 4.3 }$$

2) $\xi_u(t)$ is weakly left-continuous at every point $t\in[\tau,T]$ as an $\mathscr E$-valued function;

3) the initial conditions are satisfied in the sense that

$$\begin{equation*} \xi_u(\tau)=\{u_\tau,u'_\tau\},\quad \xi_u(\tau+0):=\operatorname{w\text{-}lim}_{t\to\tau+}\xi_u(t)= \bigl\{u_\tau,u'_\tau+\mu(\{\tau\})\bigr\}. \end{equation*}$$

As in the linear case (see Remarks 3.2 and 3.3), it can be shown that

$$\begin{equation*} \partial_t u(t+0)-\partial_t u(t)=\mu(\{t\}),\qquad t\in[\tau,T], \end{equation*}$$

and, in particular, the difference $\xi_{u_1-u_2}(t)$ between two energy solutions of (4.1) (corresponding to the same $\mu$) belongs to $C_{\mathrm{w}}(\tau,T,\mathscr{E})$. In addition, exactly as in the linear case, the equality (4.3) is equivalent to

$$\begin{equation} \begin{aligned} \, \nonumber \partial_t u(t)&=-\int_\tau^t[(-\Delta_x+1)u(s)+f(u(s))]\,ds \\ &\qquad-\gamma u(t)+\mu([\tau,t))+u_\tau'+\gamma u_\tau,\qquad t\in[\tau,T]. \end{aligned} \end{equation} \tag{ 4.4 }$$

The presence of the non-linear term $f(u)$ does not make any difference here, due to the embedding theorem $H^1\subset L^6$ and the growth assumption $f(u)\,{\in}\,L^\infty(\tau,T;H^{-1})$.

The next step is a theorem on the solvability of the problem (4.1) in the class of energy solutions.

Theorem 4.2.  Let $\xi_\tau\in\mathscr E$ and $\mu\in M_b(\mathbb{R},H)$, and let the non-linearity $f$ satisfy (4.2). Then in the sense of the above definition for every $T>\tau$ there exists at least one energy solution $u(t)$ which satisfies the estimate

$$\begin{equation} \|\xi_u(t)\|_{\mathscr{E}}\leqslant e^{-\beta(t-\tau)}Q(\|\xi_\tau\|_{\mathscr{E}})+ Q(\|\mu\|_{M_b(\tau,T;H)}),\qquad t\geqslant\tau, \end{equation} \tag{ 4.5 }$$

where the monotone increasing function $Q$ and the constant $\beta>0$ are independent of $\tau\in\mathbb{R}$, $T$, $\mu$, and $\xi_\tau\in\mathscr{E}$.

Proof.  Indeed, let us start with the case where the measure $\mu$ is regular, that is, $\mu\in L^1(\tau,T;H)$. In this case the assertion of the theorem is standard: the existence of a weak solution can be obtained, for example, using Galerkin approximations. Uniform energy estimates for the Galerkin approximations can be deduced just by multiplying the equation by $\partial_t u+\beta u$ for some positive $\beta$, and the validity of the energy estimate for the solution of (4.1) is then established by passing to the limit in the Galerkin approximations (see [12] and the references therein for details).

Consider now the general case, when the measure $\mu$ can be singular. In this case, we approximate $\mu$ by regular measures $\mu_n$ using the special approximations constructed in Proposition 9.11 (see also Remark 9.12). Namely, the sequence $\mu_n$ is uniformly bounded in $L^1(\tau,T;L^2)$ and weak-star convergent to the measure $\mu$ in $M(\tau,T;H)$, and the $\mu_n([\tau,t))$ converge to $\mu([\tau,t))$ for every $t\in[\tau,T]$. Let $u_n(t)$ be an energy solution of (4.1), with $\mu$ replaced by $\mu_n$. Then by the uniform energy estimate we may assume without loss of generality that

$$\begin{equation*} \xi_{u_n}\to \xi_{u}\quad \text{weak-star in $L^\infty(\tau,T;\mathscr{E})$}. \end{equation*}$$

Due to the compactness of the embedding

$$\begin{equation*} L^\infty(\tau,T;H^1)\cap W^{1,\infty}(\tau,T;H)\subset C(\tau,T;H) \end{equation*}$$

we conclude that $u_n\to u$ strongly in $C(\tau,T;H)$ and therefore almost everywhere in $[\tau,T]\times\Omega$. Moreover, since $f(u_n)$ is uniformly bounded in $L^{6/5}((\tau,T)\times\Omega)$, the convergence almost everywhere implies that

$$\begin{equation*} f(u_n)\to f(u) \quad \text{weakly in $L^{6/5}((\tau,T)\times\Omega)$}. \end{equation*}$$

The convergence established lets us pass to the limit as $n\to\infty$ in the equations (4.3) and prove that the limit function $u$ solves (4.1) in the sense of distributions.

Finally, in order to verify left-continuity (find a suitable representative in the equivalence class), it is sufficient to pass to the pointwise limit in equation (4.4) for the solutions $u_n$, and the theorem is proved. $\square$

The existence of weak energy solutions can be proved similarly not just for quintic non-linearities. The only difference is that the energy space should be properly defined in the case of rapidly growing non-linearities. Namely, if the non-linearity grows like $u|u|^q$ with $q>4$, then one should take $\mathscr{E}:=[H^1_0\cap L^{q+2}]\cap L^2$ as the energy space (see [12] for details). However, to the best of our knowledge, the uniqueness of such a solution is not known for $q\geqslant4$.

Moreover, for the quintic case $q=4$ we also do not know whether or not any energy solution satisfies the energy estimate (4.5). In order to overcome this problem, we introduce (following [4], [34], and [33]) the so-called Shatah–Struwe solutions and use the Strichartz estimates.

Definition 4.3.  An energy solution $u(t)$ is a Shatah–Struwe solution of the problem (4.1) if, in addition,

$$\begin{equation} u\in L^4(\tau,T;L^{12}(\Omega)). \end{equation} \tag{ 4.6 }$$

Note that, since

$$\begin{equation*} \|f(u)\|_{L^1(\tau,T;H)}\leqslant C\bigl(1+\|u\|^4_{L^4(\tau,T;L^{12})}\bigr)\|u\|_{L^\infty(\tau,T;H^1)}, \end{equation*}$$

for any Shatah–Struwe solution $u$ we have $f(u)\in L^1(\tau,T;H)$.

The next theorem establishes the uniqueness of such solutions.

Theorem 4.4.  Let $u_1$ and $u_2$ be two Shatah–Struwe solutions of the problem (4.1) which correspond to different initial data and the same measure $\mu\in M(\tau,T,H)$. Then the following estimate holds:

$$\begin{equation} \|\xi_{u_1}(t)-\xi_{u_2}(t)\|_{\mathscr{E}}\leqslant \exp\biggl\{C\int_\tau^t(1+\|u_1(s)\|^4_{L^{12}}+ \|u_2(s)\|^4_{L^{12}})\,ds\biggr\} \|\xi_{u_1}(0)-\xi_{u_2}(0)\|_{\mathscr{E}}, \end{equation} \tag{ 4.7 }$$

where $\tau\leqslant t\leqslant T$ and the constant $C$ is independent of $u_1$ and $u_2$. In particular, the Shatah–Struwe solution is unique.

Proof.  Indeed, let $v(t)=u_1(t)-u_2(t)$. Then the function $\xi_v(t)$ is weakly continuous in $\mathscr{E}$, since $\partial_t u_1$ and $\partial_t u_2$ make the same jumps determined by the discrete part $\mu_{\mathrm{d}}$. This function solves the equation

$$\begin{equation*} \partial_t^2v+\gamma\,\partial_t v+(1-\Delta_x) v+v+[f(u_1)-f(u_2)]=0. \end{equation*}$$

Since $f(u_1)-f(u_2)\in L^1([\tau,T],L^2)$ and $\partial_t v\in L^\infty([\tau,T],L^2)$, multiplication by $\partial_t v$ can be justified in the standard way and gives

$$\begin{equation*} \frac{1}{2}\,\frac d{dt}\|\xi_v\|^2_{\mathscr{E}}= -(f(u_1)-f(u_2),\partial_t v). \end{equation*}$$

Moreover, using again the estimate $|f'(u)|\leqslant C(1+|u|^4)$, the Hölder inequality, and the embedding $H^1\subset L^6$, we get that

$$\begin{equation*} \begin{aligned} \, |(f(u_1)-f(u_2),\partial_t v)|&\leqslant C((1+|u_1|^4+|u_2|^4)|v|,|\partial_t v|) \\ &\leqslant C(1+\|u_1\|_{L^{12}}^4+ \|u_1\|_{L^{12}}^4)\|v\|_{H^1}\|\partial_t v\|_{L^2}, \end{aligned} \end{equation*}$$

and the Gronwall inequality finishes the proof of the theorem. $\square$

The next corollary is crucial for our proof of asymptotic compactness.

Corollary 4.5.  Let the assumptions of Theorem 4.2 hold, and in addition let the measure $\mu$ be non-atomic (that is, $\mu(\{t\})=0$ for all $t$). Then for every Shatah– Struwe solution $u$, the energy functional $t\to \|\xi_u(t)\|^2_{\mathscr{E}}/2+(F(u(t)),1)$ is a continuous $\mathrm{BV}$-function of time and the following energy equality holds for all $\tau\leqslant s\leqslant t\leqslant T$:

$$\begin{equation} \begin{aligned} \, \nonumber &\frac{1}{2}\|\xi_u(t)\|^2_{\mathscr{E}}+(F(u(t)),1)- \frac{1}{2}\|\xi_u(s)\|^2_{\mathscr{E}}-(F(u(s)),1)+ \gamma\int_s^t\|\partial_t u(\kappa)\|^2_{L^2}\,d\kappa \\ &\qquad=\int_s^t(\partial_t u(\kappa),\mu(d\kappa)). \end{aligned} \end{equation} \tag{ 4.8 }$$

In particular, $\xi_u\in C(\tau,T;\mathscr{E})$.

Proof.  Indeed, since $u\in L^5(\tau,T;L^{10})$, the term $f(u)\in L^1(\tau,T;H)$ can be treated as a regular measure. Thus, according to Corollary 3.7, we may write

$$\begin{equation*} \begin{aligned} \, &\frac{1}{2}\bigl(\|\xi_u(t)\|^2_{\mathscr{E}}- \|\xi_u(s)\|^2_{\mathscr{E}}\bigr)+ \int_s^t\bigl(f(u(\kappa)),\partial_t u(\kappa)\bigr)\,d\kappa+ \gamma\int_s^t\|\partial_t u(\kappa)\|^2_{L^2}\,d\kappa \\ &\qquad=\int_s^t\bigl(\partial_t u(\kappa),\mu(d\kappa)\bigr). \end{aligned} \end{equation*}$$

Since $f(u)\partial_t u\in L^1((\tau,T)\times\Omega)$, the term involving the non-linearity is well-defined. Moreover, arguing in the standard way, we get that the function $t\to (F(u(t)),1)$ is absolutely continuous and

$$\begin{equation*} (F(u(t)),1)-(F(u(s)),1)=\int_s^t(f(u(\kappa)),\partial_t u(\kappa))\,d\kappa. \end{equation*}$$

Thus, the energy equality is proved. The fact that the energy functional is continuous follows immediately from this equality. Finally, the fact that $\xi_u\in C(\tau,T;\mathscr{E})$ follows from the energy equality in a straightforward way using the energy method. $\square$

We now discuss the existence of Shatah–Struwe solutions.

Proposition 4.6.  Let the assumptions of Theorem 4.2 be satisfied. Then for any $\xi_u(\tau)\in\mathscr E$ there exists a unique global Shatah–Struwe solution $u(t)$, and it satisfies the energy dissipative estimate (4.5).

The proof of existence is standard (see [35], [9], and [10] for details). First, based on the Strichartz estimate (3.16) for the linear equation and treatment of the non-linearity as a perturbation, one establishes local existence. Then by using the so-called Pohozhaev–Morawetz identity and non-concentration arguments one establishes that the Strichartz norm cannot blow up in finite time, and this gives global existence. The presence of the measure $\mu$ on the right-hand side does not produce any essential difficulties. We do not give a detailed proof here since in the next section we give an alternative proof and estimate the Strichartz norm without using non-concentration arguments.

As we have already mentioned, the global existence result for Shatah–Struwe solutions based on non-concentration arguments (and stated in Proposition 4.6) does not give any control of the Strichartz norm $\|u\|_{L^4(T,T+1;L^{12})}$ in terms of $T$ and the corresponding norms of the initial data and the external forces. In particular, we do not have any control of the behaviour of this norm as $T\to\infty$, which in turn leads to essential problems in attractor theory (see [18] for details). The aim of this section is to estimate this Strichartz norm in terms of the energy norm and a suitable norm of the external forces. Since we already know a dissipative estimate for the energy norm, this result will give us the desired dissipative estimate for the Strichartz norm. Our approach is crucially based on the following result for solutions of the homogeneous quintic wave equation in the whole of $\mathbb{R}^3$.

Proposition 5.1.  There exists a monotone increasing function $Q\colon \mathbb{R}_+\to\mathbb{R}_+$ such that any Shatah–Struwe solution $v(t)$ of the quintic wave equation

$$\begin{equation} \partial_t v-\Delta_x v+v^5=0 \end{equation} \tag{ 5.1 }$$

in the whole space $\Omega=\mathbb{R}^3$ has the estimate

$$\begin{equation} \|v\|_{L^4(\mathbb{R},L^{12}(\mathbb{R}^3))}\leqslant Q\bigl(\|\partial_t v(0)\|_{L^2(\mathbb{R}^3)}^2+ \|\nabla_x v(0)\|^2_{L^2(\mathbb{R}^3)}\bigr). \end{equation} \tag{ 5.2 }$$

The proof of this estimate can be found in [4] (see also [39] for an explicit expression for $Q$).

Clearly, the estimate (5.2) on the whole line $t\in\mathbb{R}$ cannot hold in the case where $\Omega$ is a bounded domain. However, its finite-time analogue remains true for $\Omega=\mathbb T^3$.

Corollary 5.2.  There exists a monotone increasing function $Q\colon \mathbb{R}_+\to\mathbb{R}_+$ such that any Shatah–Struwe solution $v$ of the quintic wave equation (5.1) with periodic boundary conditions has the estimate

$$\begin{equation} \|v\|_{L^4(0,1;L^{12})}\leqslant Q(\|\xi_v(0)\|_{\mathscr E}). \end{equation} \tag{ 5.3 }$$

Indeed, this estimate follows immediately from (5.2) and the result on a finite speed of propagation for wave equations (see [35]). To the best of our knowledge, the question of the validity of (5.3) in the case of general bounded domains remains open.

We are now ready to state the key result of this section.

Theorem 5.3.  Let $\Omega=\mathbb T^3$, let the non-linearity $f$ satisfy (4.2), and let the external force $\mu$ be in $M(0,1;H)$. Then the Shatah–Struwe solution $u$ of the problem

$$\begin{equation} \partial_t^2u-\Delta_x u+f(u)=\mu,\qquad \xi_u\big|_{t=0}=\xi_0, \end{equation} \tag{ 5.4 }$$

satisfies the estimate

$$\begin{equation} \|\xi_u\|_{L^\infty(0,1;\mathscr{E})}+\|u\|_{L^4(0,1;L^{12})}\leqslant Q(\|\xi_0\|_{\mathscr{E}})+Q(\|\mu\|_{M(0,1;H)}), \end{equation} \tag{ 5.5 }$$

where the monotone non-decreasing function $Q$ is independent of the choice of the initial data $\xi_0\in\mathscr{E}$ and $\mu\in M(0,1;H)$.

Proof.  We suppose first that $f(u)=u^5$ (that is, $h(u)=0$). The general case $h\ne0$ will be considered later. To verify the desired estimate, we consider a sequence $\{u_N\}_{N=1}^\infty$ approximating the solution $u$, where $u_N$ solves the problem (5.4) with the external force $\mu_N$ instead of $\mu$, and the sequence $\{\mu_N\}_{N=1}^\infty$ of discrete measures is provided by Theorem 9.21:

$$\begin{equation} \mu_N=\sum_{k=0}^{N}h_{k,N}\delta_{t_{k,N}},\quad\text{where}\quad h_{k,N}\in H,\quad \sum_{k=0}^{N}\|h_{k,N}\|\leqslant\|\mu\|_{M(0,1;H)}, \end{equation} \tag{ 5.6 }$$

and $0= t_{0,N}<t_{1,N}<\cdots<t_{N,N}=1$. Note that the solution $u_N(t)$ should solve the homogeneous problem for $t\in(t_{k,N},t_{k+1,N})$ and has jumps of the time derivative at finitely many points $t=t_{k,N}$:

$$\begin{equation*} \partial_t u_N(t_{k,N}+0)-\partial_t u_N(t_{k,N}-0)=h_{k,N}. \end{equation*}$$

Thus, the existence and uniqueness of $u_N$ follows immediately from the analogous result for the homogeneous problem (5.1) and we need not use Proposition 4.6 here. Moreover, by Theorem 4.2 we have the uniform energy estimate

$$\begin{equation} \begin{aligned} \, \|\xi_{u_N}\|_{L^\infty(0,1;\mathscr E)}&\leqslant Q(\|\xi_0\|_{\mathscr E})+Q(\|\mu_N\|_{M(0,1;H)})\nonumber\\ &\leqslant Q(\|\xi_0\|_{\mathscr E})+Q(\|\mu\|_{M(0,1;H)}) \end{aligned} \end{equation} \tag{ 5.7 }$$

for some monotone increasing function $Q$. Hence by passing to a subsequence if necessary and using the fact that $\Phi_{\mu_N}(t)\to\Phi_\mu(t)$ uniformly for all $t$ (due to the special choice of $\mu_N$ explained in Theorem 9.21), we may assume that $u_N$ is weak-star convergent to the weak energy solution $u$ of (5.4) (see the proof of Theorem 4.2). Thus, we only need to verify the uniform estimate of the Strichartz norms for the solutions $u_N$. Then passing to the limit as $N\to\infty$ will give us the desired estimate for $u$ as well.

Note that we can get the Strichartz estimate for the solution $u_N$ just by applying the estimate (5.3) on every time interval $t\in(t_{k,N},t_{k+1,N})$ and using the fact that the energy norm is under control. However, this is not enough, since the resulting estimate will clearly depend on $N$. So we need to proceed a bit more carefully.

Consider the approximations $u_N^l$, $l=0,\dots,N-1$, of the solution $u_N$ which solve (5.4) with the same initial data and with the external forces

$$\begin{equation*} \mu_N^l:=\sum_{k=0}^{l}h_{k,N}\delta_{t_{k,N}}. \end{equation*}$$

Then on the one hand,

$$\begin{equation} \|\xi_{u_N^l}\|_{L^\infty(0,1;\mathscr E)}\leqslant Q(\|\xi_0\|_{\mathscr E})+Q(\|\mu\|_{M(0,1;L^2)}) \end{equation} \tag{ 5.8 }$$

uniformly with respect to $l$ and $N$ by Theorem 4.2. On the other hand, it is clear that $u_N^{N}(t)\equiv u_N(t)$ for all $t\in[0,1]$. Moreover,

$$\begin{equation*} u_N^{l}(t)=u_N^{l+1}(t),\qquad t<t_{l+1,N}, \end{equation*}$$

and the functions $u_N^{l}(t)$ and $u_{N}^{l+1}(t)$, $t\geqslant t_{l+1,N}$, solve the linear homogeneous problem (5.1) with the initial data

$$\begin{equation*} \xi_{u_N^{l+1}}=\xi_{u_{N}^l}\big|_{t=t_{l+1,N}+0}+\{0,h_{l+1,N}\}. \end{equation*}$$

In particular, by Corollary 5.2 and the estimate (5.8),

$$\begin{equation} \|u_N^{l}\|_{L^4(t_{l,N},1;L^{12})}+\|u_N^{l+1}\|_{L^4(t_{l+1,N},1;L^{12})} \leqslant Q(\|\xi_0\|_{\mathscr E})+Q(\|\mu\|_{M(0,1;H)}). \end{equation} \tag{ 5.9 }$$

Finally, we introduce the functions $v_{0}(t):=u_N^0(t)$ and $v_{l+1}(t):=u_{N}^{l+1}(t)-u_N^l(t)$, $l=0,\dots, N-1$. Then obviously,

$$\begin{equation} u_N(t)=v_0(t)+\sum_{l=1}^{N}v_l(t),\qquad t\in[0,1], \end{equation} \tag{ 5.10 }$$

and the functions $v_{l+1}(t)$, $t>t_{l+1,N}$, satisfy the equations

$$\begin{equation} \partial_t^2 v_{l+1}-\Delta_x v_{l+1}=-[f(u_N^{l+1})-f(u_N^l)],\quad \xi_{v_{l+1}}\big|_{t=t_{l+1,N}+0}=\{0,h_{l+1,N}\}. \end{equation} \tag{ 5.11 }$$

Note also that $v_{l+1}(t)\equiv0$ for $t<t_{l+1,N}$. To estimate the Strichartz norms of the functions $v_{l+1}$, we use that

$$\begin{equation*} |f(u^{l+1}_N)-f(u^l_N)|\leqslant C(|u^{l+1}_N|^4+|u^l_N|^4)|v_{l+1}|, \end{equation*}$$

and therefore by the Hölder inequality and the Sobolev embedding $H^1\subset L^6$ we have

$$\begin{equation} \|f(u^{l+1}_N)-f(u^l_N)\|_{H}\leqslant C\bigl(\|u^{l+1}_N\|^4_{L^{12}}+ \|u^l_N\|^4_{L^{12}}\bigr)\|\xi_{v_{l+1}}\|_{\mathscr E}. \end{equation} \tag{ 5.12 }$$

Multiplying (5.11) by $\partial_t v_{l+1}$ and using (5.12), we now get that

$$\begin{equation*} \frac{1}{2}\,\frac d{dt}\|\xi_{v_{l+1}}\|^2_{\mathscr E}\leqslant C\bigl(\|u^{l+1}_N\|^4_{L^{12}}+\|u^l_N\|^4_{L^{12}}\bigr) \|\xi_{v_{l+1}}\|_{\mathscr E}^2, \end{equation*}$$

and the Gronwall inequality together with the control (5.9) gives us that

$$\begin{equation} \|\xi_{v_{l+1}}\|_{L^\infty(0,1;\mathscr{E})}\leqslant \bigl(Q(\|\xi_0\|_{\mathscr E})+Q(\|\mu\|_{M(0,1;H)})\bigr)\|h_{l+1,N}\|_{H}. \end{equation} \tag{ 5.13 }$$

We are now ready to apply the standard Strichartz estimate to the linear equation (5.11) and get that

$$\begin{equation} \begin{aligned} \, \nonumber \|v_{l+1}\|_{L^4(0,1;L^{12})}&=\|v_{l+1}\|_{L^4(t_{l+1,N},1;L^{12})} \\ \nonumber &\leqslant C\bigl(\|\xi_{v_{l+1}}(t_{l+1,N}+0)\|_{\mathscr E}+ \|f(u_N^{l+1})-f(u_N^l)\|_{L^1(t_{l+1,N},1;H)}\bigr) \\ & \leqslant \bigl(Q(\|\xi_0\|_{\mathscr E})+Q(\|\mu\|_{M(0,1;H)})\bigr) \|h_{l+1,N}\|_{H}. \end{aligned} \end{equation} \tag{ 5.14 }$$

Finally, according to (5.10),

$$\begin{equation} \begin{aligned} \, \nonumber \|u_N\|_{L^4(0,1;L^{12})}&\leqslant \bigl(Q(\|\xi_0\|_{\mathscr E})+ Q(\|\mu\|_{M(0,1;H)})\bigr)\biggl(1+\sum_{l=0}^{N}\|h_{l,N}\|_{H}\biggr) \\ &\leqslant \bigl(Q(\|\xi_0\|_{\mathscr E})+Q(\|\mu\|_{M(0,1;H)})\bigr) (1+Q(\|\mu\|_{M(0,1;H)})). \end{aligned} \end{equation} \tag{ 5.15 }$$

Thus, the theorem is proved in the particular case $f(u)=u^5$.

We now consider the general case $h(u)\ne0$, which can be derived from the estimate obtained using more or less standard perturbation theory arguments. First we recall the following simple lemma, which can be verified using convexity arguments (see [42] and also [32]).

Lemma 5.4.  Let $Q\colon \mathbb{R}_+\to \mathbb{R}_+$ be a monotone increasing function, let $L_1,L_2\in\mathbb{R}_+$, and let $\varepsilon\in[0,1]$. Then there exists a smooth monotone increasing function $Q_1\colon \mathbb{R}_+\to\mathbb{R}_+$ such that

$$\begin{equation} Q(L_1+\varepsilon L_2)\leqslant Q_1(L_1)+\varepsilon Q_1(L_2), \end{equation} \tag{ 5.16 }$$

where $Q_1$ is determined by $Q$ only.

We rewrite (5.4) in the form

$$\begin{equation*} \partial_t^2 u-\Delta_x u+u^5=\mu -h(u) \end{equation*}$$

and apply the already proved estimate (5.5) on the interval $t\in[0,T]$ where $T\leqslant1$ will be determined later. This gives us the estimate

$$\begin{equation*} \begin{aligned} \, \|u\|_{L^5(0,T;L^{10})}&\leqslant \|u\|_{L^4(0,T;L^{12})}+ \|\xi_u\|_{L^\infty(0,T;\mathscr E)} \\ &\leqslant Q(\|\xi_0\|_{\mathscr E})+Q(\|\mu\|_{M(0,1;H)}+ \|h(u)\|_{L^1(0,T;H)}). \end{aligned} \end{equation*}$$

Since the function $h(u)$ has a sub-quintic growth rate, the Hölder inequality gives us that

$$\begin{equation*} \|h(u)\|_{L^1(0,T;L^2)}\leqslant CT^\kappa\bigl(1+\|u\|_{L^5(0,T;L^{10})}^5\bigr) \end{equation*}$$

for some positive exponent $\kappa$. Inserting this estimate into the previous one and using Lemma 5.4, we get that

$$\begin{equation} \|u\|_{L^5(0,T;L^{10})}\leqslant Q(\|\xi_0\|_{\mathscr E})+ Q(\|\mu\|_{M(0,1;L^2)})+T^\kappa Q(\|u\|_{L^5(0,T;L^{10})}). \end{equation} \tag{ 5.17 }$$

Important here is that the function $Q$ is independent of $T$. Fixing $T=T(\|\xi_0\|_{\mathscr E}+\|\mu\|_{M(0,1;H)})$ to be small enough, we derive from (5.17) that

$$\begin{equation*} \|u\|_{L^5(0,T;L^{10})}\leqslant Q(\|\xi_0\|_{\mathscr E})+Q(\|\mu\|_{M(0,1;H)}) \end{equation*}$$

for some new monotone increasing function $Q$. Since the energy norm of the solution is under control, we may apply this estimate on the intervals $[T,2T]$, $[2T,3T]$, and so on. This gives us the desired control

$$\begin{equation*} \|u\|_{L^5(0,1;L^{10})}\leqslant Q(\|\xi_0\|_{\mathscr E})+Q(\|\mu\|_{M(0,1;H)}) \end{equation*}$$

with some monotone increasing function $Q$. Since the $L^1(H)$-norm of $f(u)$ is controlled by the $L^5(L^{10})$-norm of $u$, we can get control of the $L^4(L^{12})$-norm of $u$ using the Strichartz estimate for the linear equation. Thus, the theorem is proved. $\square$

As a consequence of Theorem 5.3, we obtain the desired dissipative Strichartz estimate for the solutions of the non-linear damped wave equation (1.1), which is crucial for what follows.

Corollary 5.5.  Let the non-linearity $f$ satisfy (4.2), and let the external force $\mu$ be in $M_b(\mathbb{R};H)$. Then for any $\tau\in\mathbb{R}$ and any initial data $\xi_\tau\in \mathscr{E}$ the problem (1.1) has a unique Shatah–Struwe solution $u$, and the following estimate holds: for $t\geqslant\tau$,

$$\begin{equation} \|\xi_u(t)\|_{\mathscr{E}}+\|u\|_{L^4(t,t+1;L^{12})}\leqslant e^{-\beta(t-\tau)}Q(\|\xi_\tau\|_{\mathscr{E}})+ Q(\|\mu\|_{M_b(\mathbb{R};H)}) \end{equation} \tag{ 5.18 }$$

for some constant $\beta>0$ and some monotone increasing function $Q$ which are independent of $\xi_\tau\in\mathscr{E}$, $\mu\in M_b(\mathbb{R};H)$, and $\tau\in\mathbb{R}$.

Proof.  This result easily follows if one applies the estimate (5.5) to the equation (1.1) on $[t,t+1]$, treating $\gamma\partial_t u +u$ as the right-hand side, and then combines the resulting estimate with the dissipative energy estimate (4.5) and the estimate in Lemma 5.4. $\square$

Remark 5.6.  Since the $L^4(L^{12})$-norm of the solution $u$ together with the energy norm allow us to control the $L^1(L^2)$-norm of the non-linearity $f(u)$, by applying the Strichartz estimates for the linear equation and treating $f(u)$ as an external force we get a dissipative estimate for other Strichartz norms of $u$, namely,

$$\begin{equation} \|u\|_{L^{2/q}(t,t+1;L^{6/(1-q)})}\leqslant Q(\|\xi_u(0)\|_{\mathscr E})e^{-\alpha t}+Q(\|\mu\|_{M_b(\mathbb{R},H)}), \end{equation} \tag{ 5.19 }$$

where $q\in[0,1)$ and $Q$ depends on $q$ but not on $u$ and $\mu$.

Remark 5.7.  We recall that the Strichartz estimates for non-homogeneous linear dispersive equations are usually derived from the homogeneous equations using duality arguments and the so-called Christ–Kiselev lemma (see [38] and the references therein). In contrast to this, the approach proposed in the proof of Theorem 5.3 works directly for non-linear (and even critically non-linear) problems and can be treated as a generalisation of the Christ–Kiselev lemma to the non-linear case. We believe that this approach will also be useful for other dispersive equations.

We start with basic definitions of non-autonomous dynamical systems (adapted to the measure-driven case). For a more detailed treatment and recent advances see [12] and [45]. Let us first recall the key definitions and concepts related to attractor theory, beginning with the autonomous case.

Definition 6.1.  Let $\Phi$ be a Hausdorff topological space, and let $\mathbb S(t)\colon \Phi\to\Phi$, $t\geqslant0$, be a one-parameter semigroup on it. Also, let $\mathbb B$ be a family of sets $B\subset \Phi$ with the property that if $B\in\mathbb B$ and $B_1\subset B$, then $B_1\in\mathbb B$. The sets $B\in \mathbb B$ are said to be bounded.

A set $\mathscr B\in \mathbb B$ is called an absorbing set for the semigroup $\mathbb S(t)$ if for any $B\in\mathbb B$ there exists a time $T=T(B)$ such that

$$\begin{equation*} \mathbb S(t)B\subset\mathscr B,\qquad t\geqslant T. \end{equation*}$$

A set $\mathscr B$ is called an attracting set for the semigroup $S(t)$ if for every neighbourhood $\mathscr O(\mathscr B)$ and every $B\in\mathbb B$ there exists a time $T=T(\mathscr O,B)$ such that

$$\begin{equation*} \mathbb S(t)B\subset\mathscr O(\mathscr B),\qquad t\geqslant T. \end{equation*}$$

Finally, a set $\mathscr A$ is a global attractor for the semigroup $\mathbb S(t)$ if:

1) $\mathscr A$ is compact and bounded ($\mathscr A\in\mathbb B$) in $\Phi$;

2) $\mathscr A$ is an attracting set for $\mathbb S(t)$;

3) $\mathscr A$ is a minimal set with the properties 1) and 2).

Property 3) of the global attractor is usually formulated as strict invariance with respect to $\mathbb S(t)$, but keeping in mind the non-autonomous case, we prefer to state it as minimality (see [12] for more details). To state the existence result for the autonomous case we need one more definition.

Definition 6.2.  The semigroup $\mathbb S(t)\colon \Phi\to\Phi$ is (sequentially) asymptotically compact on a set $B\subset\Phi$ if, for any sequences $t_n\to\infty$ and $x_n\in B$, the sequence $\mathbb S(t_n)x_n$ is precompact in $\Phi$.

Proposition 6.3.  Let the semigroup $\mathbb S(t)\colon \Phi\to\Phi$ possess an absorbing set $\mathscr B\in\mathbb B$. Assume also that:

1) the topology induced on $\mathscr B$ by the inclusion $\mathscr B\subset\Phi$ is metrizable and complete (that is, $\mathscr B$ is a complete metric space);

2) the semigroup $\mathbb S(t)$ is asymptotically compact on $\mathscr B$.

Then $\mathbb S(t)$ has a global attractor $\mathscr A\subset\mathscr B$.

In addition, if the operators $\mathbb S(t)$ are continuous on $\mathscr B$ for every fixed $t$, then the attractor $\mathscr A$ is strictly invariant: $\mathbb S(t)\mathscr A=\mathscr A$, and is generated by all bounded trajectories defined for all $t\in\mathbb{R}$:

$$\begin{equation} \mathscr A=\mathscr K\big|_{t=0}, \end{equation} \tag{ 6.1 }$$

where $\mathscr K:=\bigl\{u\colon\mathbb{R}\to\Phi\colon \mathbb S(t)u(\tau)= u(t+\tau),\ t\geqslant0,\ \tau\in\mathbb{R},\ \bigcup_{t\in\mathbb{R}}u(t)\in\mathbb B\bigr\}$.

The proof of this proposition is standard and the details can be found in [12].

Since we are mainly interested in the non-autonomous equations, we recall below how the above concepts and results can be extended to the non-autonomous case. The first difference is that the solution operators form not a semigroup but a so-called dynamical process, which is a two-parameter family $U(t,\tau)$, $t\geqslant\tau$, acting in the phase space and satisfying the relations

$$\begin{equation} U(t,t)=\operatorname{Id},\qquad U(t,\tau)\circ U(\tau,s)=U(t,s),\quad t\geqslant\tau\geqslant s. \end{equation} \tag{ 6.2 }$$

The operator $U(t,\tau)$ is understood as a solution operator which maps the initial data at the time $\tau$ to the solution at the time $t$.

Definition 6.4.  Let $\mathscr E$ be a Hausdorff topological space, and let $U(t,\tau)\colon \mathscr E\to\mathscr E$, $t\geqslant\tau$, be a dynamical process on it. Also, let $\mathbb B$ be a family of sets $B\subset \mathscr E$ such that if $B\in\mathbb B$ and $B_1\subset B$, then $B_1\in\mathbb B$. The sets $B\in \mathbb B$ are said to be bounded.

A set $\mathscr B\in \mathbb B$ is called a uniformly absorbing set for the process $U(t,\tau)$ if for any $B\in\mathbb B$ there exists a time $T=T(B)$ such that

$$\begin{equation*} U(t,\tau)B\subset\mathscr B,\qquad t-\tau\geqslant T,\quad \tau\in\mathbb{R}. \end{equation*}$$

A set $\mathscr B$ is called a uniformly attracting set for the process $U(t,\tau)$ if for every neighbourhood $\mathscr O(\mathscr B)$ and every $B\in\mathbb B$, there exists a time $T=T(\mathscr O,B)$ such that

$$\begin{equation*} U(t,\tau)B\subset\mathscr O(\mathscr B),\qquad t-\tau\geqslant T,\quad \tau\in\mathbb{R}. \end{equation*}$$

Finally, a set $\mathscr A_{\mathrm{un}}$ is called a uniform attractor for the process $U(t,\tau)$ if:

• 1)  
  $\mathscr A$ is compact and bounded in $\mathscr E$;
• 2)  
  $\mathscr A$ is a uniformly attracting set for $U(t,\tau)$;
• 3)  
  $\mathscr A$ is a minimal set with the properties 1) and 2).

Below, $\mathscr E$ will usually be a Banach space (or even a Hilbert space) endowed with either the strong or the weak topology. The associated uniform attractor will be referred to as a strong or a weak unform attractor, respectively. In both cases $\mathbb B$ consists of all bounded sets in the Banach space under consideration.

Generalisation of the concept of asymptotic compactness to the non-autonomous case is also straightforward.

Definition 6.5.  A process $U(t,\tau)\colon \mathscr E\to\mathscr E$ is uniformly asymptotically compact on a set $B\subset\mathscr E$ if, for any sequences $t_n,\tau_n\in\mathbb{R}$ such that $t_n-\tau_n\to\infty$ and any sequence $x_n\in B$, the sequence $U(t_n,\tau_n)x_n$ is precompact in $\mathscr E$.

The existence theorem is generalised in a similar way (see [12] for details).

Proposition 6.6.  Let the process $U(t,\tau)\colon \mathscr E\to\mathscr E$ possess a uniformly absorbing set $\mathscr B\in\mathbb B$. Assume also that:

1) the topology induced on $\mathscr B$ by the inclusion $\mathscr B\subset\Phi$ is metrizable and complete (that is, $\mathscr B$ is a complete metric space);

2) the process $U(t,\tau)$ is uniformly asymptotically compact on $\mathscr B$.

Then $U(t,\tau)$ has a uniform attractor $\mathscr A_{\mathrm{un}}\subset\mathscr B$.

We now return to our damped wave equation (1.1). Since, according to Corollary 5.5, for any $\tau\in\mathbb{R}$, $\xi_\tau\in\mathscr E:=H^1(\mathbb T^3)\times L^2(\mathbb T^3)$, and $\mu\in M_b(\mathbb{R},H)$ this problem has a unique Shatah–Struwe solution $\xi_u(t)$, we can introduce a family of dynamical processes $U_\mu(t,\tau)$, $\mu\in M_b(\mathbb{R},H)$, in the energy phase space. However, since in contrast to the usual case, the trajectories $\xi_u(t)$ may have jumps, we should be a bit more careful in order to preserve the property (6.2). In particular, here we use our convention that the trajectories $\xi_u(t)$ are left-continuous. Therefore, we may let

$$\begin{equation} U_\mu(t,\tau)\xi_\tau:=\xi_u(t-0)=\lim_{s\to t-0}\xi_u(s)=\xi_u(t), \end{equation} \tag{ 6.3 }$$

where $\xi_u(\tau-0):=\xi_\tau$. Then it is not difficult to see that the operators $U_\mu(t,\tau)$ thus defined are indeed dynamical processes in the energy space $\mathscr E$, so we can study their uniform attractors. We fix $\mathbb B$ as the family of all bounded (in the usual sense) subsets of our energy space $\mathscr E$ (it is a Banach space, hence bounded sets are well defined). Then the estimate (5.18) guarantees the existence of a uniformly absorbing set. Moreover, it can be taken in the form of a ball

$$\begin{equation} \mathscr B:=\{\xi\in\mathscr E\colon \|\xi\|_{\mathscr E}\leqslant 2Q(2\|\mu\|_{M_b(\mathbb{R},H)})\}. \end{equation} \tag{ 6.4 }$$

Recall that in this section we are mainly interested in weak uniform attractors, so we endow the space $\mathscr E$ with the weak topology and denote the resulting locally convex space by $\mathscr E_w$. Since $\mathscr E$ is a reflexive Banach space, the absorbing set $\mathscr B$ is compact and metrizable in the weak topology of $\mathscr E_w$, hence all the assumptions of Proposition 6.6 are automatically satisfied, and we have proved the following result.

Theorem 6.7.  Let the assumptions of Corollary 5.5 hold. Then for every $\mu\in M_b(\mathbb{R},H)$ the dynamical process $U_\mu(t,\tau)$ has a uniform attractor $\mathscr A_{\mathrm{un}}^w$ in $\mathscr E_w$, which is called a weak uniform attractor for the equation (1.1).

As the next step we describe the extension of the key representation formula (6.1) to the case of uniform attractors. To this end we use (following [12]) the reduction of the dynamical process $U_\mu(t,\tau)$ to a semigroup acting on an extended phase space. Namely, we introduce the group of shifts acting on the space of measures $M_b(\mathbb{R},H)$ according to the formula

$$\begin{equation*} (T(s)\mu)(t):=\mu(t+s),\qquad t,s\in\mathbb{R}. \end{equation*}$$

Then it is not difficult to verify that the above dynamical process $U_\mu(t,s)$ satisfies the translation identity (that is, the cocycle property)

$$\begin{equation} U_{T(s)\mu}(t,\tau)=U_\mu(t+s,\tau+s), \qquad t\geqslant\tau\in\mathbb{R},\quad s\in\mathbb{R}. \end{equation} \tag{ 6.5 }$$

In order to fix a suitable topology on the space $M_b(\mathbb{R},H)$, we recall that $M_{\mathrm{loc}}(\mathbb{R}, H)$ is the dual space of $C_{00}(\mathbb{R},H)$, where $C_{00}$ means the space of continuous functions with compact support endowed with the inductive limit topology. Denote by $M_{\mathrm{loc}}^{w^*}(\mathbb{R},H)$ the space $M_{\mathrm{loc}}(\mathbb{R},H)$ endowed with the associated weak-star topology. Then by the Banach– Alaoglu theorem the unit ball of $M_b(\mathbb{R},H)$ is precompact and metrizable in the topology of $M_{\mathrm{loc}}^{w^*}(\mathbb{R},H)$. We recall that $\mu_n\to\mu$ in this topology if and only if

$$\begin{equation*} \lim_{n\to\infty}\,\int_\mathbb{R}(\phi(s),\mu_n(ds))= \int_\mathbb{R}(\phi(s),\mu(ds)) \end{equation*}$$

for any $\phi\in C_{00}(\mathbb{R},H)$. We are now ready to define the hull of the measure $\mu\in M_b(\mathbb{R},H)$ to be the closure in the weak-star topology of the set of all shifts of $\mu$:

$$\begin{equation} \mathscr H(\mu)= [\{T(s)\mu,s\in\mathbb{R}\}]_{M_{\rm loc}^{w^*}(\mathbb{R},H)}, \end{equation} \tag{ 6.6 }$$

where $[\,\cdot\,]_{M_{\mathrm{loc}}^{w^*}(\mathbb{R},H)}$ means the closure in $M_{\mathrm{loc}}^{w^*}(\mathbb{R},H)$. Obviously, the set $\mathscr H(\mu)$ endowed with the weak-star topology is a compact metric space and the group of shifts

$$\begin{equation*} T(s)\colon\mathscr H(\mu)\to \mathscr H(\mu),\qquad T(s)\mathscr H(\mu)=\mathscr H(\mu), \end{equation*}$$

acts continuously on $\mathscr H(\mu)$.

Remark 6.8.  Note that, in contrast to the usual case, the norm in the space $M_b(\mathbb{R},H)$ is not lower semicontinuous in the induced weak-star topology, that is, there exist sequences $\mu_n$ bounded in $M_b(\mathbb{R},H)$ such that $\mu_n\rightharpoondown\mu$ weak-star in $M_{\mathrm{loc}}^{w^*}(\mathbb{R},H)$ but

$$\begin{equation*} \|\mu\|_{M_b(\mathbb{R},H)}> \liminf_{n\to\infty}\|\mu_n\|_{M_b(\mathbb{R},H)}. \end{equation*}$$

For this reason the unit ball in $M_b(\mathbb{R},H)$ is not closed in $M_{\mathrm{loc}}^{w^*}(\mathbb{R},H)$. On the other hand, from (9.28) it is not difficult to prove that

$$\begin{equation} \|\mu\|_{M_b(\mathbb{R},H)}\leqslant 2\liminf_{n\to\infty}\|\mu_n\|_{M_b(\mathbb{R},H)}, \end{equation} \tag{ 6.7 }$$

which is enough for our purposes. In particular, the weak-star closure of the unit ball of $M_b(\mathbb{R},H)$ in $M_{\mathrm{loc}}^{w^*}(\mathbb{R},H)$ is a subset of the ball in $M_b(\mathbb{R},H)$ of radius $2$. Note also that the constant $2$ is sharp. Indeed, consider the sequence of measures $\mu_n:=\delta_{-1/n}+\delta_{1+1/n}$. Then it is obvious that $\mu_n$ is weak-star convergent to $\mu:=\delta_0+\delta_1$, but

$$\begin{equation*} \|\mu_n\|_{M_b(\mathbb{R},H)}=1,\quad \|\mu\|_{M_b(\mathbb{R},H)}=2. \end{equation*}$$

Now let $U_\mu(t,\tau)\colon \mathscr E\to\mathscr E$ be a family of dynamical processes associated with the damped wave equation (1.1) with right-hand sides $z\in\mathscr H(\mu)$. Then the extended phase space for the problem (1.1) is defined by

$$\begin{equation*} \Phi:=\mathscr E\times\mathscr H(\mu), \end{equation*}$$

and the associated semigroup on $\Phi$ acts as follows:

$$\begin{equation} \mathbb S(t)\{\xi_0,z\}:=\{U_z(t,0),T(t)z\},\qquad \xi_0\in \mathscr E,\quad z\in\mathscr H(\mu). \end{equation} \tag{ 6.8 }$$

Indeed, the semigroup property for $\mathbb S(t)$ is an immediate consequence of the translation identity (6.5).

The key general idea is to relate the above uniform attractor $\mathscr A_{\mathrm{un}}$ for the dynamical process $U_\mu(t,\tau)$ to the global attractor $\mathbb A$ of the extended semigroup $\mathbb S(t)$ and, in particular, to describe the structure of $\mathscr A_{\mathrm{un}}$ using the representation (6.1) for the autonomous case. Namely, we endow the extended phase space $\Phi=\mathscr E\times\mathscr H(\mu)$ with the topology induced by the embedding $\Phi\subset \mathscr E_w\times M_{\mathrm{loc}}^{w^*}(\mathbb{R},H)$, and we fix bounded sets in $\Phi$ as follows: $B\subset \Phi$ is bounded if and only if $\Pi_1B$ is bounded in $\mathscr E$ (here and below, $\Pi_1$ is the projection onto the first component of the direct product $\mathscr E\times\mathscr H(\mu)$). Then by the estimate (5.18) and the inequality

$$\begin{equation} \|z\|_{M_b(\mathbb{R},H)}\leqslant 2\|\mu\|_{M_b(\mathbb{R},H)},\qquad z\in \mathscr H(\mu), \end{equation} \tag{ 6.9 }$$

the set

$$\begin{equation*} \mathscr B_{\rm ext}:=\mathscr B\times\mathscr H(\mu) \end{equation*}$$

with $\mathscr B$ defined by (6.4) is a compact metrizable absorbing set for the extended semigroup $\mathbb S(t)$, and therefore by Proposition 6.3 the semigroup $\mathbb S(t)$ possesses a global attractor $\mathbb A_{\mathrm{ext}}$. The next theorem gives the desired description of the structure of the uniform attractor for the damped wave equation (1.1), under some extra continuity assumptions.

Theorem 6.9.  Let the assumptions of Theorem 6.7 hold, and in addition let the maps $(\xi_0,z)\to U_z(t,\tau)\xi_0$ be continuous (in the weak topology) as maps from $\mathscr B_{\mathrm{ext}}$ to $\mathscr{E}$ for all $t,\tau\in\mathbb{R}$ with $t\geqslant\tau$. Then

$$\begin{equation} \mathscr A_{\rm un}=\Pi_1\mathbb A_{\rm ext} \end{equation} \tag{ 6.10 }$$

and, moreover,

$$\begin{equation} \mathscr A_{\rm un}=\bigcup_{z\in\mathscr H(\mu)}\mathscr K_z\big|_{t=0}, \end{equation} \tag{ 6.11 }$$

where $\mathscr K_z:=\{u\in L^\infty(\mathbb{R},\mathscr E)\colon U_z(t,\tau)u(\tau)=u(t), \ t\geqslant\tau\in\mathbb{R}\}$ is the so-called kernel of the process $U_z(t,\tau)$ in the terminology of [12].

The proof of this result in a general setting can be found in [12].

Note that, in contrast to the usual case, the continuity assumption is not satisfied for general measures $\mu\in M_b(\mathbb{R},H)$. Namely, the following result holds.

Proposition 6.10.  Let the assumptions of Theorem 6.7 hold. Then the continuity assumption of Theorem 6.9 holds if and only if

$$\begin{equation} 1)\quad z_n(\{\tau\})\equiv 0,\qquad 2)\quad z_n([\tau,t])\to z([\tau,t]) \quad\textit{weakly in $H$} \end{equation} \tag{ 6.12 }$$

for every sequence $z_n\in\mathscr H(\mu)$ such that $z_n\to z$ weak-star in $M_{\mathrm{loc}}(\mathbb{R},H)$ and for every fixed $t\geqslant\tau\in\mathbb{R}$.

Proof.  Indeed, let (6.12) be satisfied. We need to prove that $U_{z_n}(t,\tau)\xi_n$ converges weakly to $U_z(t,\tau)\xi_0$ as $z_n\to z$ in $\mathscr H(\mu)$ and $\xi_n\to\xi_0$ in $\mathscr E_w$. Let $\xi_{u_n}(t):=U_{z_n}(t,\tau)\xi_n$ be the corresponding Shatah–Struwe solutions. Then by the uniform dissipative estimate (5.18) we may assume without loss of generality that $\xi_{u_n}\to\xi_u$ weak-star in $L^\infty(\tau,t;\mathscr E)$. Thus, we only need to pass to the limit in (4.4). Namely, taking into account that $z_n(\{t\})=0$, we see that this equality is

$$\begin{equation} \partial_t u_n(t)=-\int_\tau^t[(-\Delta_x+1)u_n(s)+f(u_n(s))]\,ds- \gamma u_n(t)+z_n([\tau,t])+u_{\tau,n}'+\gamma u_{\tau,n}, \end{equation} \tag{ 6.13 }$$

where $\xi_n:=\{u_{\tau,n},u_{\tau,n}'\}$. Obviously, the limit function $\xi_u(t)$ satisfies (1.1) in the sense of distributions, and the passage to the limit in (6.13) is also straightforward due to the condition (6.12).

Let us now check necessity. We first check the necessity of the condition that $z(\{\tau\})=0$ for all $z\in\mathscr H(\mu)$. Indeed, let $z(\{0\})\ne0$ for some $z\in\mathscr H(\mu)$. Since the number of jumps is at most countable, we may assume that $z(\{-1\})=0$. Consider the sequence $z_n:=T_{1/n}z$ and let $\xi_{u_n}:=U_{z_n}(-1,0)\xi_0$, where $\xi_0\in\mathscr E$. Clearly, $z_n\to z$ as $n\to\infty$ and we may assume without loss of generality that $\xi_{u_n}\to\xi_{\overline u}$ weak-star in $L^\infty(-1,0;\mathscr E)$. Moreover, by the Helly selection theorem, we may also assume that $\xi_{u_n}(t)\to\xi_{\overline u}(t)$ weakly in $\mathscr E$ for almost all $t\in[-1,0]$. Let $\xi_u(t):=U_{z}(-1,t)\xi_0$. Then two cases are a priori possible.

1) $\xi_{\overline u}(t)\ne\xi_{u}(t)$ on a subset of $[-1,0]$ of positive measure, and continuity obviously fails.

2) $\xi_{\overline u}=\xi_{u}$ almost everywhere. In this case by passing to the limit in (6.13), say, in $H^{-2}$, we get that

$$\begin{equation} \begin{aligned} \, \nonumber &\lim_{n\to\infty}\partial_t u_n(0-)-\partial_t u(0-)= \lim_{n\to\infty} z_n([-1,0))-z([-1,0)) \\ &\quad=-\lim_{n\to\infty}z\biggl(\biggl[-1,-1+\frac{1}{n}\biggr)\biggr)+ \lim_{n\to\infty}z\biggl(\biggl[-1,\frac{1}{n}\biggr)\biggr)- z([-1,0))=z(\{0\})\ne0, \end{aligned} \end{equation} \tag{ 6.14 }$$

and the continuity of $U_z(-1,0)$ also fails.

Thus, the necessity of the first condition is proved.

The necessity of the second condition can be proved similarly, but even more simply, since we need not shift measures and we can pass to the limit directly in (6.13). $\square$

This proposition reduces finding necessary and sufficient conditions for weak continuity of the dynamical process associated with (1.1) to verifying the conditions (6.12), which are purely measure theoretic and can be completely analysed. To formulate the definitive criterion we need the following definition.

Definition 6.11.  A measure $\mu\in M_b(\mathbb{R},H)$ is weakly uniformly non-atomic if for every $\psi\in H$ there exists a monotone increasing function $\omega_\psi\colon \mathbb{R}_+\to\mathbb{R}_+$ such that

$$\begin{equation} \lim_{h\to0}\omega_\psi(h)=0\quad\text{and}\quad |(\mu([s,t]),\psi)|\leqslant \omega_\psi(|t-s|) \end{equation} \tag{ 6.15 }$$

for all $t\geqslant s\in\mathbb{R}$. The space of such measures is denoted by $M^{\mathrm{wna}}_b(\mathbb{R},H)$.

Then the following result holds.

Proposition 6.12.  The conditions (6.12) are satisfied if and only if the initial measure $\mu\in M_b(\mathbb{R},H)$ is weakly uniformly non-atomic.

Proof.  Assume that the conditions (6.12) hold and let $\psi\in H$ be arbitrary. Consider the function $G\colon \mathscr H(\mu)\times[0,1]\to\mathbb{R}$ defined by

$$\begin{equation*} G(z,\tau):=(z([0,\tau]),\psi). \end{equation*}$$

Then in view of the first condition in (6.12) this function is continuous in $\tau$ for any fixed $z$. On the other hand, by the second condition in (6.12) it is continuous in $z$ for any fixed $\tau$. Thus, there is a point $\tau_0\in(0,1)$ such that $G$ is jointly continuous at $\{z,\tau_0\}$ for any $z\in\mathscr H(\mu)$ (in fact, there is a dense set of such points $\tau_0\in[0,1]$; see [31] and the references therein, for example). Since $\mathscr H(\mu)$ is compact, we conclude that there exists a monotone increasing function $\omega_\psi\colon \mathbb{R}_+\to\mathbb{R}_+$ such that

$$\begin{equation*} |(z([\tau_0,\tau_0+s]),\psi)|=|G(z,\tau_0+s)-G(z,\tau_0)|\leqslant \omega_\psi(s) \quad \forall\, z\in\mathscr H(\mu) \end{equation*}$$

and $\lim_{h\to0}\omega_\psi(h)=0$. Finally, using the equalities

$$\begin{equation*} (T(h)z)([\tau,t])=z([\tau+h,t+h]) \end{equation*}$$

and $T(h)\mathscr H(\mu)=\mathscr H(\mu)$, we deduce (6.15). Thus, the conditions (6.12) imply that $\mu$ is weakly uniformly non-atomic.

Now let $\mu$ be weakly uniformly non-atomic. Then it is not difficult to see using the Helly selection theorem (see Theorem 9.13 and Corollary 9.18) that

$$\begin{equation} |(z([\tau,t]),\psi)|\leqslant\omega_\psi(t-\tau)\quad \forall\,z\in\mathscr H(\mu), \end{equation} \tag{ 6.16 }$$

where the functions $\omega_\psi$ are the same as in (6.15). Then the first assumption of (6.12) is immediate, and the second is a standard corollary of the Arzelà theorem. $\square$

Thus, we have proved the following theorem, which can be considered as the main result of this section.

Theorem 6.13.  Let the assumptions of Theorem 6.7 hold, and in addition let $\mu\in M^{\mathrm{wna}}_b(\mathbb{R},H)$. Then the weak uniform attractor $\mathscr A_{\mathrm{un}}$ of equation (1.1) satisfies (6.10) and (6.11).

Indeed, this is an immediate consequence of Theorem 6.9 and Propositions 6.10 and 6.12.

We now give some examples clarifying the conditions imposed on the external force.

Example 6.14.  We start with the case of regular measures $\mu(t)\in L^p_b(\mathbb{R},H)$, where $p>1$. Then

$$\begin{equation*} \begin{aligned} \, \|\mu([\tau,\tau+h])\|_H&=\biggl\|\int_\tau^{\tau+h}\mu(t)\,dt\biggr\|_H \leqslant\int_\tau^{\tau+h}\|\mu(t)\|_H\,dt \\ &\leqslant\biggl(\int_\tau^{\tau+h}1\,dt\biggr)^{1-1/p}\|\mu\|_{L^p_b}= C|h|^{1-1/p}. \end{aligned} \end{equation*}$$

Thus, $\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H)$ (and is even strongly uniformly non-atomic) and the above theory works. Moreover, in this case

$$\begin{equation*} \mathscr H(\mu)\subset L^p_b(\mathbb{R},H)\subset L^1_b(\mathbb{R},H), \end{equation*}$$

so that all measures in the hull are regular.

This will not be the case if we consider so-called normal external forces in $L^1_b(\mathbb{R},H)$, which were introduced in [27] to study uniform attractors for parabolic equations (see also [45] for more details). We recall that $\mu\in L^1_b(\mathbb{R},H)$ is normal if there is a monotone increasing function $\omega\colon \mathbb{R}_+\to\mathbb{R}_+$ such that $\lim_{h\to0}\omega(h)=0$ and

$$\begin{equation} \int_t^{t+h}\|\mu(t)\|_H\,dt\leqslant \omega(h),\qquad t\in\mathbb{R}. \end{equation} \tag{ 6.17 }$$

In this case we still have $\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H)$ (and even $\mu\in M_b^{\mathrm{sna}}(\mathbb{R},H)$), and the theory works. However, in this case the hull $\mathscr H(\mu)$ may contain measures with non-zero singular part. According to the Dunford–Pettis theorem (see §9) the condition ensuring the embedding $\mathscr H(\mu)\subset L^1_b(\mathbb{R},H)$ is a bit stronger:

$$\begin{equation*} \int_A\|\mu(t)\|_H\,dt\leqslant \omega(|A|), \end{equation*}$$

where $A$ is any (Lebesgue) measurable set in $\mathbb{R}$ and $|A|$ is its Lebesgue measure.

The condition (6.17) can be weakened as follows:

$$\begin{equation} \biggl\|\int_t^{t+h}\mu(t)\,dt\biggr\|_H\leqslant \omega(h),\qquad t\in\mathbb{R}, \end{equation} \tag{ 6.18 }$$

which still ensures that $\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H)$.

Example 6.15.  We now give two more exotic examples clarifying the nature of weakly non-atomic measures. We start with a scalar measure $\mu\in M_b(\mathbb{R},\mathbb{R})$. To this end we fix a non-negative smooth function $\phi\in C_0^\infty(\mathbb{R})$ supported on $[0,1]$ such that $\int_\mathbb{R}\phi(t)\,dt=1$, and we consider the delta-like sequence $\phi_n(t):=n\phi(nt)$. Finally, we introduce the function

$$\begin{equation} \mu(t):=\sum_{n=2}^\infty\biggl(\frac{1}{n} \sum_{k=1}^n (-1)^k\phi_{n^2}\biggl(t-n-\frac{k}{n^2}\biggr)\biggr). \end{equation} \tag{ 6.19 }$$

Clearly, this function belongs to $L^1_b(\mathbb{R})$. It is also not difficult to show that the $n$th term of this function averages to zero. Thus, in particular, $\mu\in M_b^{\mathrm{wna}}(\mathbb{R})$ and

$$\begin{equation*} T(s)\mu\rightharpoondown 0 \end{equation*}$$

as $s\to\infty$. On the other hand, the total variation of this measure is

$$\begin{equation*} |\mu|(t):= \sum_{n=2}^\infty\biggl(\frac{1}{n}\sum_{k=1}^n \phi_{n^2} \biggl(t-n-\frac{k}{n^2}\biggr)\biggr), \end{equation*}$$

and we see that the $n$th term here tends to the $\delta$-function at $t=n$. In particular,

$$\begin{equation*} T(n)|\mu|\rightharpoondown\sum_{n\in\mathbb Z}\delta(t-n)\ne0. \end{equation*}$$

Thus, $|\mu|\notin M_b^{\mathrm{wna}}(\mathbb{R})$, so that the assumption (6.18) does not imply (6.17), and the class of measures $M_b^{\mathrm{wna}}(\mathbb{R})$ is indeed larger than $M_b^{\mathrm{sna}}(\mathbb{R})$.

The next example is somehow complementary to the previous one and gives an alternative construction in the infinite-dimensional space. Namely, let $H$ be a Hilbert space and let $\{e_n\}_{n=1}^\infty$ be an orthonormal basis in it. Let

$$\begin{equation*} \mu(t):=\sum_{n=1}^\infty\phi_n(t-n)e_n. \end{equation*}$$

Then clearly $\mu\in L^1_b(\mathbb{R},H)$, and its total variation is

$$\begin{equation*} |\mu|(t):=\sum_{n=1}^\infty\phi_n(t-n). \end{equation*}$$

Thus, taking any $\psi\in H$ and using the fact that $(\psi,e_n)\to0$, we see that $\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H)$. However, its total variation clearly does not belong to this space.

Our last example shows the pathology which may appear in the case where the condition $\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H)$ fails.

Example 6.16.  Consider the first-order ordinary differential equation

$$\begin{equation} y'=y-y^3-3+3\tanh t. \end{equation} \tag{ 6.20 }$$

An example for the hyperbolic equation can be obtained similarly by adding the term $\varepsilon y''(t)$, but the construction becomes less transparent, so we prefer to deal with the first-order equation. In this case the uniform attractor can be found explicitly. Namely, the external force now is $\mu_0(t)=3\tanh t$, and its hull is

$$\begin{equation*} \mathscr H(\mu_0)=\{-3\}\cup\{+3\}\cup\{\mu_0(t+h),\ h\in\mathbb{R}\}. \end{equation*}$$

Moreover, using the comparison principle, for example, it is not difficult to see that every complete trajectory $y(t)$ corresponding to an external force $\mu\in \mathscr H(\mu_0)$ with $\mu\ne\pm3$ satisfies the conditions

$$\begin{equation*} \lim_{t\to-\infty}y(t)=-2,\quad \lim_{t\to+\infty}y(t)=-1,\quad y'(t)>0, \end{equation*}$$

and hence

$$\begin{equation*} \mathscr K_z=[-2,-1],\qquad z\in\mathscr H(\mu_0),\quad z\ne\pm3. \end{equation*}$$

Finally, in the case $z=-3$ the equation is monotone, hence $\mathscr K_z=\{-2\}$, and in the case $z=+3$ we have the autonomous regular attractor $\mathscr K_z=[-1,1]$. Therefore,

$$\begin{equation*} \mathscr A_{\rm un}=\bigcup_{z\in\mathscr H(\mu_0)}\mathscr K_z=[-2,1]. \end{equation*}$$

We now consider a perturbed version of equation (6.20):

$$\begin{equation} \begin{gathered} \, y'=y-y^3-3+3\tanh t+\overline\mu(t), \\ \overline\mu(t):=\frac{1}{2}\sum_{n=1}^\infty\biggl(\phi_{Kn}(t-Kn)-\phi_{Kn} \biggl(t-Kn-\frac{1}{Kn}\biggr)\biggr), \end{gathered} \end{equation} \tag{ 6.21 }$$

where $K>0$ is a sufficiently large number and $\phi_n(t)$ is the same as in the previous example. Then since $T(s)\overline\mu\to0$ as $s\to\pm\infty$ in the weak-star topology, the hull of the external force $\mu+\overline\mu$ is similar to the non-perturbed case:

$$\begin{equation*} \mathscr H(\mu+\overline\mu)=\{+3\}\cup\{-3\}\cup\{\mu(t+h)+ \overline\mu(t+h),h\in\mathbb{R}\}. \end{equation*}$$

Then, using the fact that the term $(1/2)(\phi_{Kn}(t-Kn)-\phi_{Kn}(t-Kn-1/(Kn)))$ on the right-hand side of (6.21) just generates in the solution of (6.20) a spike of height close to $1/2$ centred near $t=Kn$ if $K$ is large enough, we see that

$$\begin{equation*} \mathscr K_z=\biggl[-2,-\frac{1}{2}\biggr],\qquad z\in\mathscr H(\mu+\overline\mu),\quad z\ne\pm3. \end{equation*}$$

Thus,

$$\begin{equation*} \bigcup_{z\in\mathscr H(\mu+\overline\mu)}\mathscr K_z=[-2,1]. \end{equation*}$$

On the other hand, if we take $y\big|_{t=\tau}=1$ with $\tau>0$ large enough, then we get a trajectory which is close to $y(t)=1$ but with spikes of height close to $1/2$. This shows that

$$\begin{equation*} \mathscr A_{\rm un}=\biggl[-2,\frac{3}{2}\biggr]\ne \bigcup_{z\in\mathscr H(\mu+\overline\mu)}\mathscr K_z. \end{equation*}$$

Remark 6.17.  We recall that the representation formula (6.10) plays a fundamental role in the theory of non-autonomous attractors (see [12], for example), so the last example shows that the theory of uniform attractors for general measures $\mu\in M_b(\mathbb{R},H)$ which we have developed is not entirely satisfactory, and we really need the restriction $\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H)$ to have a reasonable theory.

So far, the problem of constructing a satisfactory attractor theory for general measures $\mu\in M_b(\mathbb{R},H)$ remains open. The most natural and straightforward idea here is to endow the space $M_b(\mathbb{R},H)$ with a different topology in which the operators $U_\mu(t,\tau)$ would be continuous with respect to $\mu$. But unfortunately this does not work even in the scalar case. Indeed, we actually need a topology $\Upsilon$ on the space of measures $M(0,1)$ that has two properties:

1) the unit ball in $M(0,1)$ is sequentially compact in $\Upsilon$;

2) convergence $\mu_n\to\mu$ in $\Upsilon$ implies pointwise convergence $\Phi_{\mu_n}(t)\to\Phi_\mu(t)$ of the distribution functions for any $t\in[0,1]$.

But such a topology does not exist! Indeed, consider the sequence $\mu_n=\delta(t-1/2)-\delta(t-1/2-1/n)$. It clearly converges to zero in the weak-star topology and does not converge to zero in $\Upsilon$ (since $\Phi_{\mu_n}(1/2)=1$ does not converge to zero). Note that convergence in $\Upsilon$ plus uniform boundedness of a sequence implies its weak-star convergence (by the Helly theorem). Thus, we should have a subsequence $\mu_{n_k}$ which converges in $\Upsilon$ to zero, which is impossible since $\Phi_{\mu_{n_k}}(1/2)=1$ does not tend to zero. This shows that the problem is deeper than one might expect.

Alternatively, it seems to us that the problem can be solved by passing from a dynamical process on the initial phase space to a so-called trajectory dynamical system which acts on pieces of trajectories endowed with a suitable space-time topology (for example, the weak topology of $L^p_{\mathrm{loc}}(\mathbb{R}_+,\mathscr E)$ with $1\leqslant p<\infty$; see [12] and the references therein). We return to this problem in a forthcoming paper.

In this section we would like to address the question of the existence of a strong uniform attractor $\mathscr A_{\mathrm{un}}^s$ for equation (1.1). By definition, this is a uniform attractor for the dynamical process $U_\mu(t,\tau)$ associated with this equation and acting in the energy phase space $\mathscr E$ endowed with the strong topology (see Definition 6.4). In this section we always assume that

$$\begin{equation} \mu\in M_b^{\rm wna}(\mathbb{R},H), \end{equation} \tag{ 7.1 }$$

and therefore the weak uniform attractor $\mathscr A_{\mathrm{un}}^w$ always exists and has the description (6.11) by Theorem 6.13. It is also not difficult to see that the strong uniform attractor, if it exists, coincides with the weak one:

$$\begin{equation} \mathscr A_{\rm un}^s=\mathscr A_{\rm un}^w= \bigcup_{z\in\mathscr H(\mu)}\mathscr K_z. \end{equation} \tag{ 7.2 }$$

Moreover, in view of Proposition 6.6 the existence of a strong uniform attractor follows from the asymptotic compactness of the process $U_\mu(t,\tau)$. In fact, it is more convenient for us to verify instead the asymptotic compactness of the extended semigroup $\mathbb S(t)\colon \Phi\to\Phi$ acting on the space $\Phi:=\mathscr E\times\mathscr H(\mu)$, where the space $\mathscr E$ is endowed with the strong topology (and $\mathscr H(\mu)$ remains endowed with the weak-star topology). Namely, we will verify that for any sequence $\tau_n\in\mathbb{R}$ such that $\tau_n\to-\infty$ and any sequences $z_n\in\mathscr H(\mu)$ and $\xi_{\tau_n}\in \mathscr B$, the sequence

$$\begin{equation} \{U_{z_n}(0,\tau_n)\xi_{\tau_n}\}_{n=1}^\infty \end{equation} \tag{ 7.3 }$$

is precompact in $\mathscr E$. By the translation identity, this implies the asymptotic compactness of the process $U_\mu(t,\tau)$. Actually, since under our conditions the extended semigroup $\mathbb S(t)$ is weakly continuous on $\Phi$ for every fixed $t\geqslant0$, one can prove that the asymptotic compactness of the semigroup $\mathbb S(t)$ and that of the process $U_\mu(t,\tau)$ are equivalent, but we will not use this fact below.

Clearly, the assumption $\mu\in M^{\mathrm{wna}}_b(\mathbb{R},H)$ alone is not enough to get strong asymptotic compactness (see examples in [45]). In particular, as shown in [45], $\mu\in L^\infty(\mathbb{R},H)$ is also not enough for compactness even in the case of a linear damped wave equation. In order to state our extra assumptions on $\mu$, we use the following classes of external forces introduced in [45].

Definition 7.1.  Let $\mu\in M_b(\mathbb{R},H)$. The measure $\mu$ is said to be space-regular if there exists a sequence $\mu_n\in M_b(\mathbb{R},C^\infty(\Omega))$ such that

$$\begin{equation} \lim_{n\to\infty}\|\mu_n-\mu\|_{M_b(\mathbb{R},H)}=0. \end{equation} \tag{ 7.4 }$$

Similarly, $\mu$ is said to be time-regular if there exists a sequence $\mu_n\in C^\infty_b(\mathbb{R},H)$ such that (7.4) holds (here and below we identify a measure which is absolutely continuous with respect to Lebesgue measure with its density).

The following proposition gives the key property of the classes of measures introduced.

Proposition 7.2.  Let $\mu\in M_b(\mathbb{R},H)$ be space-regular. Then for any $k\in \mathbb{N}$ and any $\varepsilon>0$ there exists a $\overline\mu=\overline\mu_{\varepsilon,k}\in M_b(\mathbb{R},H^k)$ such that

$$\begin{equation} \|\mu-\overline\mu\|_{M_b(\mathbb{R},H)}\leqslant\varepsilon. \end{equation} \tag{ 7.5 }$$

Moreover, every measure in $\mathscr H(\mu)$ is space-regular, and for every $z\in\mathscr H(\mu)$ there exists a $\overline z\in\mathscr H(\overline\mu)$ such that

$$\begin{equation} \|z-\overline z\|_{M_b(\mathbb{R},H)}\leqslant\varepsilon. \end{equation} \tag{ 7.6 }$$

Similarly, let $\mu\in M_b(\mathbb{R},H)$ be time-regular. Then for any $k\in \mathbb N$ and any $\varepsilon>0$ there exists a $\overline\mu=\overline\mu_{\varepsilon,k}\in H^k_b(\mathbb{R},H)$ such that (7.5) holds. Moreover, every measure in $\mathscr H(\mu)$ is time-regular and for every $z\in\mathscr H(\mu)$ there exists a $\overline z\in\mathscr H(\overline\mu)$ such that (7.6) holds.

The proof of this proposition is straightforward and is given in [45].

Remark 7.3.  More details on the properties of space- or time-regular functions can be found in [45]. For instance, any time-regular measure $\mu$ belongs to $L^1_b(\mathbb{R},H)$ (this follows, for example, from the Dunford–Pettis theorem; see Theorem 9.20). In contrast to this, space-regular measures may have discrete and singular components. It is also known that $\mu$ is simultaneously space- and time-regular if and only if it is translation-compact in $L^1_b(\mathbb{R},H)$.

The typical examples of space- or time-regular measures are $\mu\in M_b(\mathbb{R},H^1)$ or $\mu\in C^\alpha_b(\mathbb{R},H)$, $\alpha>0$, respectively. A typical example of a measure that is not space-regular is

$$\begin{equation*} \mu(t)=\sum_{n=1}^\infty\chi_{[n,n+1)}(t)e_n, \end{equation*}$$

where $\{e_n\}_{n=1}^\infty$ is an orthonormal basis in $H$, say, generated by the Laplacian, and $\chi_A(t)$ is the characteristic function of a set $A$. An example of a measure that is not time-regular is even simpler: $\mu(t)=\sin t^2$. Combining these two examples, we get a measure

$$\begin{equation*} \widetilde \mu(t)=\sum_{n=1}^\infty\sin(n^2t)\chi_{[n,n+1)}(t)e_n \end{equation*}$$

that is neither space- nor time-regular. Nevertheless, $\widetilde\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H)$, and as elementary calculations show, this inclusion ensures strong asymptotic compactness due to time averaging. Thus, the conditions introduced are not necessary for asymptotic compactness. Unfortunately, necessary and sufficient conditions are not yet known.

We are now ready to state and prove the main result of this section.

Theorem 7.4.  Let the assumptions of Theorem 6.13 hold, and in addition let the external force $\mu$ be time-regular or space-regular. Then the dynamical processes $U_\mu(t,\tau)$ associated with the problem (1.1) have a strong uniform attractor $\mathscr A_{\mathrm{un}}^s$, which coincides with the weak attractor $\mathscr A_{\mathrm{un}}^w$ constructed above and admits the representations (6.10) and (6.11).

Proof.  As explained before, we only need to verify the asymptotic compactness of the associated process $U_\mu(t,\tau)$ in the strong topology of $\mathscr E$. To this end it is sufficient to verify the precompactness of the sequence (7.3), where $\tau_n\to-\infty$, the $\xi_{\tau_n}$ are taken in a uniformly absorbing set $\mathscr B$, and $z_n\in\mathscr H(\mu)$. To verify this we will use the so-called energy method (see [5], [30]), which is based on the following elementary property: if $\xi_n\rightharpoondown\xi_\infty$ in the Hilbert space $\mathscr E$ and $\|\xi_n\|_\mathscr E\to\|\xi_\infty\|_{\mathscr E}$, then $\xi_n\to\xi_\infty$ strongly. The proof is divided into two natural steps.

Step 1. In this step we use the weak continuity of the processes $U_z(t,\tau)$ and the existence of a weak uniform attractor to obtain a good description of weak limit points of the sequence (7.3). The arguments given below actually re-prove the general representation formula (6.11) in the case of equation (1.1). Nevertheless, we decided to give these arguments here since they are crucial for our proof of asymptotic compactness.

Without loss of generality we may assume that $z_n\to z\in\mathscr H(\mu)$ (in the associated weak-star topology). Let us also introduce the solutions which correspond to this sequence:

$$\begin{equation} \xi_{u_n}(t)=U_{z_n}(t,\tau_n)\xi_{\tau_n},\qquad t\geqslant\tau_n. \end{equation} \tag{ 7.7 }$$

Then by the dissipative estimate (5.18), the inequality (6.9), and the fact that the $\xi_{\tau_n}$ are uniformly bounded, the sequence $\xi_{u_n}(t)$ satisfies the estimate

$$\begin{equation} \|\xi_{u_n}(t)\|_{\mathscr E}+\|u_n\|_{L^4(t,t+1;L^{12})}\leqslant C,\qquad t\geqslant\tau_n. \end{equation} \tag{ 7.8 }$$

In particular, the sequence (7.3) is bounded, so that by passing to a subsequence if necessary we may assume that

$$\begin{equation} \xi_n:=U_{z_n}(0,\tau_n)\xi_{\tau_n}\rightharpoondown \xi_\infty \end{equation} \tag{ 7.9 }$$

for some $\xi_\infty\in\mathscr E$. Moreover, without loss of generality, we may also assume that

$$\begin{equation} \xi_{u_n}\to\xi_u\quad \text{weak-star in $L^\infty_{\rm loc}(\mathbb{R},\mathscr E)$}\quad\text{and}\quad u_n\to u\quad \text{weakly in $L^4_{\rm loc}(\mathbb{R},L^{12})$} \end{equation} \tag{ 7.10 }$$

for some function $u$ such that $\xi_u\in L^\infty(\mathbb{R},\mathscr E)$ and $u\in L^4_b(\mathbb{R},L^{12})$. Passing to the limit as $n\to\infty$ in the sense of distributions in the equations (1.1) for $u_n$, we get in a standard way (for example, see [18] for details) that $u$ is a complete bounded solution of (1.1) with right-hand side $z\in\mathscr H(\mu)$, and since $z\in M_b^{\mathrm{wna}}(\mathbb{R},H)$, the function $\xi_u(t)$ has no jumps, so that $u(t)$ is a Shatah–Struwe solution of (1.1). Therefore,

$$\begin{equation*} \xi_u\in \mathscr K_z. \end{equation*}$$

We need to check now that $\xi_u(0)=\xi_\infty$. To this end we establish some results on strong convergence of $u_n(t)$ which will be essential in Step 2 below. First we note that $u_n$ is bounded in $L^\infty(\mathbb{R},H^1)$ and $\partial_t u_n$ is bounded in $L^\infty(\mathbb{R},H)$, so by compactness arguments,

$$\begin{equation} u_n\to u \quad \text{strongly in $C_{\rm loc}(\mathbb{R},H)$}. \end{equation} \tag{ 7.11 }$$

The analogous result for $\partial_t u_n(t)$ is a bit more delicate since, in contrast to the standard case, the $\partial_t^2 u_n$ are not functions but measures. To overcome this problem, we deduce from (6.13) that

$$\begin{equation} \begin{aligned} \, \nonumber \|\partial_t u_n(t)-\partial_t u_n(\tau)\|_{H^{-1}}&\leqslant \int_\tau^t(\|u_n(s)\|_{H^1}+\|f(u_n(s))\|_{H^{-1}})\,ds \\ \nonumber &\qquad+\gamma\|u_n(t)-u_n(\tau)\|_{H^{-1}}+ \|z_n([\tau,t])\|_{H^{-1}} \\ &\leqslant C|t-\tau|+\|z_n([\tau,t])\|_{H^{-1}}, \end{aligned} \end{equation} \tag{ 7.12 }$$

where we have implicitly used that $\xi_{u_n}(t)$ is bounded in $\mathscr E$ and that

$$\begin{equation*} \|f(u_n)\|_{H^{-1}}\leqslant C\|f(u_n)\|_{L^{6/5}}\leqslant C\bigl(1+\|u_n\|^6_{L^6}\bigr)\leqslant C\bigl(1+\|u_n\|_{H^1}^6\bigr). \end{equation*}$$

Moreover, since $\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H)$, there exists a monotone increasing function $\omega\colon \mathbb{R}_+\to\mathbb{R}_+$ such that $\lim_{x\to0}\omega(x)=0$ and

$$\begin{equation} \|z([\tau,t])\|_{H^{-1}}\leqslant \omega(|t-\tau|),\qquad z\in\mathscr H(\mu). \end{equation} \tag{ 7.13 }$$

Thus,

$$\begin{equation*} \|\partial_t u_n(t)-\partial_t u_n(\tau)\|_{H^{-1}}\leqslant C|t-\tau|+\omega(|t-\tau|), \end{equation*}$$

and the functions $\partial_t u_n(t)$ are equicontinuous as functions with values in $H^{-1}$. Since they are also uniformly bounded as functions with values in $H$, the Arzelà theorem gives us that

$$\begin{equation} \partial_t u_n\to\partial_t u \quad \text{strongly in $C_{\rm loc}(\mathbb{R},H^{-1})$}. \end{equation} \tag{ 7.14 }$$

Thus, $\xi_{u_n}\to \xi_u$ strongly in $C_{\mathrm{loc}}(\mathbb{R},\mathscr E^{-1})$ and, in particular,

$$\begin{equation} \xi_n=\xi_{u_n}(0)=U_{z_n}(0,\tau_n)\xi_{\tau_n}\rightharpoondown \xi_u(0)=\xi_\infty. \end{equation} \tag{ 7.15 }$$

Step 2. In this step we verify that $\|\xi_{u_n}(0)\|_{\mathscr E}\to \|\xi_u(0)\|_{\mathscr E}$ by passing to the limit in the corresponding energy equality. Crucial for this method is the fact that if $\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H)$, then any Shatah–Struwe solution of equation (1.1) satisfies the energy equality (see Corollary 4.5). Thus, the validity of taking the scalar product of the equation (1.1) with $\partial_t u$ is justified, and testing the equation with $u$ does not require any extra justification. For this reason we may multiply the equation (1.1) for the solution $u_n(t)$ by $\partial_t u_n+\delta u_n$ (following [18]), where $\delta>0$ is sufficiently small, to get that

$$\begin{equation} {\frac{d}{dt}\mathscr{E}(\xi_{u_n})+\delta\mathscr{E}(\xi_{u_n})+B(\xi_{u_n})+ \delta[(f(u_n),u_n)-(F(u_n),1)]=(z_n,\partial_t u_n+\delta u_n)}, \end{equation} \tag{ 7.16 }$$

where

$$\begin{equation} \begin{gathered} \, \mathscr{E}(\xi_u)=\frac{1}{2}\|\xi_{u}\|^2_{\mathscr{E}}+ \frac{\delta\gamma}{2}\|u\|^2_H+\delta(\partial_t u, u)+(F(u),1),\\ F(u)=\int_0^uf(v)\,dv, \end{gathered} \end{equation} \tag{ 7.17 }$$

and

$$\begin{equation} B(\xi_u)=\biggl(\gamma-\frac{3\delta}{2}\biggr)\|\partial_t u\|^2_H- \delta^2(\partial_t u,u)+\biggl(\frac{\delta}{2}- \frac{\gamma\delta^2}{2}\biggr)\|u\|^2_H+\frac{\delta}{2}\|\nabla u\|^2_H. \end{equation} \tag{ 7.18 }$$

Multiplying (7.16) by $e^{\delta t}$ and integrating the resulting identity with respect to time from $\tau_n$ to $0$, we get the energy identity in the integral form

$$\begin{equation} \begin{aligned} \, \nonumber &\mathscr{E}(\xi_{u_n})(0)+\int_{-\infty}^{0}e^{\delta s} \bigl[B(\xi_{u_n})(s)+\delta[(f(u_n(s)),u_n(s))-(F(u_n(s)),1)]\bigr]\,ds \\ &\qquad=e^{\delta\tau_n}\mathscr{E}(\xi_{\tau_n})+ \int_{-\infty}^0e^{\delta s}(\partial_t u_n(s),z_n(ds))+ \delta \int_{-\infty}^0e^{\delta s}(u_n(s),z_n(ds)), \end{aligned} \end{equation} \tag{ 7.19 }$$

where, to avoid dependence on $n$ in the lower limit of integration, we set $\xi_{u_n}(s)\equiv 0$ for $s<\tau_n$.

We want to pass to the limit as $n\to\infty$ in (7.19). To this end, we first note that the weak convergence $\xi_{u_n}(0)\to\xi_{u}(0)$ in $\mathscr E$ and the compactness of the embedding $H^1\subset H$ imply that

$$\begin{equation} \frac{\delta\gamma}{2}\|u(0)\|^2_H+\delta(\partial_t u(0),u(0))= \lim_{n\to\infty}\biggl(\frac{\delta\gamma}{2}\|u_n(0)\|^2_H+ \delta(\partial_t u_n(0), u_n(0))\biggr). \end{equation} \tag{ 7.20 }$$

In order to pass to the limit in the terms containing the non-linearity, we recall that $f(u)$ has a positive coefficient in front of the leading quintic term (see (4.2)). Therefore,

$$\begin{equation} F(s)\geqslant -C,\quad s\in\mathbb{R},\quad\text{and}\quad f(s)s-F(s)\geqslant -C,\quad s\in\mathbb{R}, \end{equation} \tag{ 7.21 }$$

for some $C=C_f$. Moreover, the strong convergence $u_n(0)\to u(0)$ in $L^2$ implies convergence almost everywhere (passing to a subsequence if necessary). This allows us to apply the Fatou lemma and get

$$\begin{equation} (F(u(0)),1)\leqslant\liminf_{n\to\infty}(F(u_n(0)),1). \end{equation} \tag{ 7.22 }$$

Similarly, using the strong convergence $u_n\to u$ in $C_{\mathrm{loc}}(\mathbb{R},H)$ and the boundedness of $u_n$ in $L^\infty(\mathbb{R},H^1)$, we arrive at the inequality

$$\begin{equation} \begin{aligned} \, \nonumber &\int_{-\infty}^0e^{\delta s}[(f(u(s)),u(s))-(F(u(s)),1)]\,ds \\ &\qquad\leqslant\liminf_{n\to\infty}\int_{-\infty}^0e^{\delta s} [(f(u_n(s)),u_n(s))-(F(u_n(s)),1)]\,ds. \end{aligned} \end{equation} \tag{ 7.23 }$$

Next, for sufficiently small $\delta=\delta(\gamma)>0$ the quadratic form $B$ is positive-definite and hence is convex and weakly lower semicontinuous. Therefore,

$$\begin{equation} \int_{-\infty}^0e^{\delta s}B(\xi_u(s))\,ds\leqslant \liminf_{n\to\infty}\int_{-\infty}^0e^{\delta s}B(\xi_{u_n}(s))\,ds. \end{equation} \tag{ 7.24 }$$

Let us now look at the right-hand side of (7.19). Since the $\xi_{\tau_n}$ are bounded in $\mathscr{E}$ by assumption and $\tau_n$ tends to $-\infty$, the first term on the right-hand side of (7.19) vanishes.

Moreover, since the $z_n$ and $u_n$ are bounded in $M_b(\mathbb{R},H)$ and $L^\infty(\mathbb{R},H)$, respectively, and $u_n\to u$ strongly in $C_{\mathrm{loc}}(\mathbb{R},H)$, we have

$$\begin{equation} \int_{-\infty}^0e^{\delta s}(u(s),z(ds))= \lim_{n\to\infty}\int_{-\infty}^0e^{\delta s}(u_n(s),z_n(ds)). \end{equation} \tag{ 7.25 }$$

Here we have also used the fact that $z_n\to z$ weak-star in $M_{\mathrm{loc}}(\mathbb{R},H)$ as well as that $\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H)$ (in order to guarantee that $z_n\big|_{t\leqslant0}\to z\big|_{t\leqslant0}$ weak-star in $M_{\mathrm{loc}}(-\infty,0;H)$).

Up to this point, we have nowhere used that $\mu$ is time- or space-regular. This will be essentially used to pass to the limit in the second term on the right-hand side of (7.19), namely, to show that

$$\begin{equation} \int_{-\infty}^0e^{\delta s}(\partial_t u(s),z(ds))= \lim_{n\to\infty}\int_{-\infty}^0e^{\delta s}(\partial_t u_n(s),z_n(ds)). \end{equation} \tag{ 7.26 }$$

We assume for the moment that (7.26) has been verified and complete the proof of the theorem. Indeed, passing to a subsequence if necessary, we may assume that

$$\begin{equation*} \limsup_{n\to\infty}\|\xi_{u_n}(0)\|_{\mathscr E}= \lim_{n\to\infty}\|\xi_{u_n}(0)\|_{\mathscr E}. \end{equation*}$$

Then by taking $\liminf_{n\to\infty}$ on both sides of (7.19) and using the inequalities obtained above together with the fact that

$$\begin{equation*} \liminf_{n\to\infty}(A_n+B_n)\geqslant \liminf_{n\to\infty}A_n+\liminf_{n\to\infty}B_n, \end{equation*}$$

we arrive at the estimate

$$\begin{equation} \begin{aligned} \, \nonumber &\limsup_{n\to\infty}\frac{1}{2}\|\xi_{u_n}(0)\|^2_{\mathscr{E}}+ \frac{\delta\gamma}{2}\|u(0)\|^2+\delta(\partial_t u(0),u(0))+(F(u(0)),1) \\ \nonumber &\qquad\qquad+\int_{-\infty}^{0}e^{\delta s}\bigl[B(\xi_{u})(s)+ \delta[(f(u(s)),u(s))-(F(u(s)),1)]\bigr]\,ds \\ &\qquad\leqslant \int_{-\infty}^0e^{\delta s}(\partial_t u(s)+ \delta u(s),z(ds)). \end{aligned} \end{equation} \tag{ 7.27 }$$

On the other hand, since $u$ is a Shatah–Struwe solution of the limit problem, it also obeys the energy equality

$$\begin{equation} \begin{aligned} \, \nonumber &\frac{1}{2}\|\xi_{u}(0)\|^2_{\mathscr{E}}+\frac{\delta\gamma}{2}\|u(0)\|^2+ \delta(\partial_t u(0),u(0))+(F(u(0)),1) \\ \nonumber &\qquad\qquad+\int_{-\infty}^{0}e^{\delta s}\bigl[B(\xi_{u})(s)+ \delta[(f(u(s)),u(s))-(F(u(s)),1)]\bigr]\,ds \\ &\qquad=\int_{-\infty}^0e^{\delta s}(\partial_t u(s)+\delta u(s),z(ds)). \end{aligned} \end{equation} \tag{ 7.28 }$$

Combining (7.27) and (7.28) with the weak lower semicontinuity of the norm $\|\cdot\|_\mathscr{E}$, we get the chain of inequalities

$$\begin{equation} \limsup_{n\to\infty}\|\xi_{u_n}(0)\|^2_{\mathscr{E}}\leqslant \|\xi_{u}(0)\|^2_{\mathscr{E}}\leqslant \liminf_{n\to\infty}\|\xi_{u_n}(0)\|^2_{\mathscr{E}}, \end{equation} \tag{ 7.29 }$$

which implies the equality

$$\begin{equation} \|\xi_{u}(0)\|^2_{\mathscr{E}}= \lim_{n\to\infty}\|\xi_{u_n}(0)\|^2_{\mathscr{E}}. \end{equation} \tag{ 7.30 }$$

Together with the already proved weak convergence $\xi_{u_n}(0)\rightharpoondown\xi_u(0)$, this proves strong convergence. Thus, to finish the proof of theorem, we only need to verify the identity (7.26). This is done in the following lemma.

Lemma 7.5.  Let $\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H)$ be a measure which is either time- or space- regular. Assume also that the sequence of functions $\xi_{u_n}\in C_b(\mathbb{R},\mathscr E)$ is uniformly bounded and that $\xi_{u_n}\to\xi_u$ strongly in $C_{\mathrm{loc}}(\mathbb{R},\mathscr E^{-1})$. Then the equality (7.26) holds for every sequence $z_n\in\mathscr H(\mu)$ such that $z_n\to z$ weak-star in $M_{\mathrm{loc}}(\mathbb{R},H)$.

Proof.  Let $\mu$ be time-regular. Then according to Proposition 7.2, for any $\varepsilon>0$ there exist $\overline\mu\in H^2_b(\mathbb{R}, H)$ and measures $\overline z_n\in \mathscr H(\overline\mu)$ such that

$$\begin{equation} \|\mu-\overline\mu\|_{M_b(\mathbb{R},H)}\leqslant\varepsilon\quad\text{and}\quad \|z_n-\overline z_n\|_{M_b(\mathbb{R},H)}\leqslant\varepsilon. \end{equation} \tag{ 7.31 }$$

Moreover, since the hull $\mathscr H(\overline\mu)$ is compact in the weak topology of $H^2_{\mathrm{loc}}(\mathbb{R},H)$, we may also assume that $\overline z_n\rightharpoondown \overline z\in\mathscr H(\overline\mu)$ weakly in $H^2_{\mathrm{loc}}(\mathbb{R},H)$. In particular,

$$\begin{equation*} \|z-\overline z\|_{M_b(\mathbb{R},H)}\leqslant\varepsilon. \end{equation*}$$

Since the functions $\xi_{u_n}(t)$ are bounded in $L^\infty(\mathbb{R},\mathscr E)$, we have

$$\begin{equation} \begin{aligned} \, \nonumber &\biggl|\int_{-\infty}^0e^{\delta s}(\partial_t u_n(s),z_n(ds))- \int_{-\infty}^0e^{\delta s}(\partial_t u_n(s),\overline z_n(ds))\biggr| \\ &\qquad+\biggl|\int_{-\infty}^0e^{\delta s}(\partial_t u(s),z(ds))- \int_{-\infty}^0e^{\delta s}(\partial_t u(s),\overline z(ds))\biggr| \leqslant C\varepsilon. \end{aligned} \end{equation} \tag{ 7.32 }$$

Thus, we only need to prove that

$$\begin{equation} \int_{-\infty}^0e^{\delta s}(\partial_t u(s),\overline z(ds))= \lim_{n\to\infty}\int_{-\infty}^0e^{\delta s} (\partial_t u_n(s),\overline z_n(ds)). \end{equation} \tag{ 7.33 }$$

To verify this we use the fact that $\overline z_n$ is smooth in time and that $u_n\to u$ strongly in $C_{\mathrm{loc}}(\mathbb{R},H)$, so we may integrate by parts and get that

$$\begin{equation} \begin{aligned} \, \nonumber &\int_{-\infty}^0e^{\delta s}(\partial_t u_n(s),\overline z_n(ds))= (\overline z_n(0),u_n(0))-\int_{-\infty}^0e^{\delta s}(\overline z_n'(s)+ \delta \overline z_n(s),u_n(s))\,ds \\ &\qquad\to (\overline z(0),u(0))- \int_{-\infty}^0 e^{\delta s}(\overline z'(s)+ \delta \overline z(s),u(s))\,ds= \int_{-\infty}^0e^{\delta s}(\partial_t u(s),\overline z(ds)). \end{aligned} \end{equation} \tag{ 7.34 }$$

This proves the lemma in the case where $\mu$ is time-regular.

Assume now that $\mu$ is space-regular. Then as in the time-regular case we may approximate the measure $\mu$ by measures $\overline \mu\in M_b(\mathbb{R},H^1)$ and fix a sequence $\overline z_n\in \mathscr H(\overline\mu)$ such that (7.31) and (7.32) hold. As before, the desired convergence will be proved if we verify (7.33). However, since we do not assume that $\overline\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H^1)$, this convergence may fail, and we need to proceed more carefully. Namely, let $\beta>0$ be a small number and let

$$\begin{equation} \theta_\beta(t):=\begin{cases} 1,& t\leqslant0, \\ 0,& t\geqslant\beta, \\ 1-\beta^{-1}t,& t\in[0,\beta]. \end{cases} \end{equation} \tag{ 7.35 }$$

Since $\overline z_n\to\overline z$ weak-star in $M_{\mathrm{loc}}(\mathbb{R},H^1)$ and $\partial_t u_n\to\partial_t u$ strongly in $C_{\mathrm{loc}}(\mathbb{R},H^{-1})$, for any $\beta>0$ we have

$$\begin{equation} \int_{\mathbb{R}}e^{\delta s}(\theta_\beta(s)\partial_t u(s),\overline z(ds))= \lim_{n\to\infty}\int_{\mathbb{R}}e^{\delta s} (\theta_\beta(s)\partial_t u_n(s),\overline z_n(ds)). \end{equation} \tag{ 7.36 }$$

Thus, to prove convergence we need to estimate

$$\begin{equation} \begin{aligned} \, \nonumber &\biggl|\int_{[0,\beta]}e^{\delta s}[(\theta_\beta(s)\partial_t u_n(s), \overline z_n(ds))-(\theta_\beta(s)\partial_t u(s),\overline z(ds))]\biggr| \\ \nonumber &\qquad\leqslant \|\overline z_n\|_{M_b(\mathbb{R},H^1)}e^{\delta\beta} \|\partial_t u_n-\partial_t u\|_{C(0,\beta;H^{-1})} \\ &\qquad\qquad+\biggl|\int_{[0,\beta]}e^{\delta s} (\theta_\beta(s)\partial_t u(s),\overline z_n(ds)-\overline z(ds))\biggr|. \end{aligned} \end{equation} \tag{ 7.37 }$$

The first term on the right-hand side tends to zero as $n\to\infty$, and by (7.31) the second term satisfies the inequality

$$\begin{equation*} \begin{aligned} \, &\biggl|\int_{[0,\beta]}e^{\delta s}(\theta_\beta(s) \partial_t u(s),\overline z_n(ds)-\overline z(ds))\biggr| \\ &\qquad\leqslant \biggl|\int_{[0,\beta]}e^{\delta s} (\theta_\beta(s)\partial_t u(s),z_n(ds)-z(ds))\biggr|+C\varepsilon, \end{aligned} \end{equation*}$$

where the constant $C$ is independent of $n$. Thus, we only need to prove that

$$\begin{equation} \lim_{\beta\to0}\int_0^\beta e^{\delta s} (\theta_\beta(s)\partial_t u(s),z(ds))=0 \end{equation} \tag{ 7.38 }$$

uniformly with respect to all $z\in\mathscr H(\mu)$. Moreover, since $z$ is non-atomic, the function $\partial_t u(s)$ is continuous as a function with values in $H$, and we only need to prove that

$$\begin{equation*} \lim_{\beta\to0}\biggl(\partial_t u(0), \int_0^\beta(1-\beta^{-1}s)\,z(ds)\biggr)=0. \end{equation*}$$

Finally, integration by parts together with the fact that $\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H)$ gives us that

$$\begin{equation*} \begin{aligned} \, \biggl|\biggl(\partial_t u(0),\int_0^\beta(1-\beta^{-1}s)\,z(ds)\biggr)\biggr|&= \biggl|\beta^{-1}\int_0^\beta(\Phi_z(s),\partial_t u(0))\,ds\biggr| \\ &\leqslant\sup_{s\in[0,\beta]}|(z([0,s]),\partial_t u(0))|\leqslant \omega_{\partial_t u(0)}(\beta). \end{aligned} \end{equation*}$$

Thus, the convergence (7.38) is verified and the lemma is proved, together with Theorem 7.4. $\square$

The aim of this section is to verify that the uniform attractor $\mathscr A_{\mathrm{un}}$ of the damped wave equation (1.1) is more regular if the external force $\mu\in M_b(\mathbb{R},H)$ is more regular. We consider two model cases of additional regularity for $\mu$, namely,

$$\begin{equation} \partial_t\mu\in M_b(\mathbb{R},H) \end{equation} \tag{ 8.1 }$$

and

$$\begin{equation} \mu\in M_b(\mathbb{R},H^\alpha) \end{equation} \tag{ 8.2 }$$

for some (small) positive $\alpha$. The main result of this section is the following theorem.

Theorem 8.1.  Let the assumptions of Corollary 5.5 hold, and in addition let the measure $\mu$ satisfy (8.1) or (8.2). Then the dynamical process $U_\mu(t,\tau)$ associated with equation (1.1) has a strong uniform attractor $\mathscr A_{\mathrm{un}}^s$ in the phase space $\mathscr E$ (which coincides with the weak uniform attractor $\mathscr A_{\mathrm{un}}^w$ constructed in Theorem 6.7), and this attractor is bounded in the space $\mathscr E^\alpha:=H^{\alpha+1}\times H^{\alpha}$ for some small $\alpha>0$:

$$\begin{equation} \|\mathscr A_{\rm un}\|_{\mathscr E^\alpha}\leqslant C. \end{equation} \tag{ 8.3 }$$

Remark 8.2.  Note that (8.1) together with the assumption that $\mu\in M_b(\mathbb{R},H)$ implies that $\mu$ is a function of bounded variation with values in $H$:

$$\begin{equation*} \mu\in \mathrm{BV}_b(\mathbb{R},H). \end{equation*}$$

In particular, $\mu\in M_b^{\mathrm{wna}}(\mathbb{R},H)$ and therefore the uniform attractor has the representations (6.10) and (6.11). In contrast to this, when (8.2) holds, the measure $\mu$ may contain a non-zero discrete part and (6.11) is not necessarily satisfied.

To prove the theorem, we split the solution $u$ into three parts

$$\begin{equation} u(t)=\theta(t)+v(t)+w(t), \end{equation} \tag{ 8.4 }$$

where $\theta(t)$ solves the linear wave equation

$$\begin{equation} \partial_t^2\theta+\gamma\,\partial_t\theta+(1-\Delta_x)\theta=\mu,\quad \xi_\theta\big|_{t=\tau}=0, \end{equation} \tag{ 8.5 }$$

the function $v$ solves the auxiliary non-linear problem

$$\begin{equation} \partial_t^2 v+\gamma\,\partial_t v+(1-\Delta_x)v+f(v)+Lv=0,\quad \xi_v\big|_{t=\tau}=\xi_u\big|_{t=\tau}, \end{equation} \tag{ 8.6 }$$

where $L>0$ is a sufficiently large number, and the remainder $w$ solves the following problem with zero initial conditions:

$$\begin{equation} \partial_t^2w+\gamma\,\partial_t w+(1-\Delta_x)w+[f(\theta+v+w)-f(v)]=Lv,\quad \xi_w\big|_{t=\tau}=0. \end{equation} \tag{ 8.7 }$$

We need to obtain good estimates for each of the three functions $\theta$, $v$, and $w$. We start with the simplest case of $\theta$, which satisfies the linear equation.

Lemma 8.3.  Let the above conditions hold and let $\mu$ satisfy either (8.1) or (8.2). Then the solution $\theta$ of (8.5) satisfies the estimate

$$\begin{equation} \|\xi_{\theta}(t)\|_{\mathscr E^\alpha}+ \|\theta\|_{L^4([t,t+1],H^{\alpha,12})}\leqslant C\|\mu\|_W, \end{equation} \tag{ 8.8 }$$

where $\alpha>0$ is sufficiently small and the symbol $W$ denotes the space $\mathrm{BV}_b(\mathbb{R},H)$ (if (8.1) holds) or $M_b(\mathbb{R},H^\alpha)$ (if (8.2) holds).

Proof.  Indeed, in the case of (8.2) the estimate (8.8) is an immediate consequence of Theorem 3.5 and the estimate (3.16) applied to the function $\overline\theta:=(-\Delta_x+1)^{\alpha/2}\theta$ and also of elliptic regularity.

Now let (8.1) be satisfied. Then by differentiating (8.5) with respect to time and writing $\overline\theta:=\partial_t\theta$, we get that

$$\begin{equation*} \partial_t^2\overline\theta+\gamma\,\partial_t\overline\theta- (\Delta_x-1)\overline\theta=\partial_t\mu,\quad \xi_{\overline\theta}\big|_{t=\tau}=\{0,\mu(\tau)\}. \end{equation*}$$

Since $\mu(\tau)$ is well defined and $\|\mu(\tau)\|_{H}\leqslant C\|\mu\|_W$, we can apply Theorem 3.5 and the estimate (3.16) to this equation and get that

$$\begin{equation*} \|\xi_{\overline\theta}(t)\|_{\mathscr E}+ \|\overline\theta\|_{L^4([t,t+1],L^{12})}\leqslant C\|\mu\|_W,\qquad t\geqslant\tau. \end{equation*}$$

Now since $(-\Delta_x+1)\theta=-\partial_t\overline\theta(t)-\gamma\overline\theta+\mu(t)$, we get that the function $\xi_{\theta}(t)$ is bounded in $\mathscr E^1$. Finally, using the fact that $H^2\subset H^{3/4,12}$, we see that (8.8) is satisfied at least for $\alpha\leqslant3/4$. Of course, the restriction $\alpha\leqslant3/4$ is artificial and can be easily removed, but the validity of (8.8) for some small positive $\alpha$ is enough for our purposes. $\square$

In the next step we show that the function $v(t)$ decays exponentially as $t\to\infty$.

Lemma 8.4.  Let the above assumptions hold. Then the solution $v(t)$ satisfies the estimate

$$\begin{equation} \|\xi_v(t)\|_{\mathscr E}+\|v\|_{L^4(t,t+1;L^{12})}\leqslant Q(\|\xi_u(\tau)\|_{\mathscr E})e^{-\delta(t-\tau)},\qquad t\geqslant\tau, \end{equation} \tag{ 8.9 }$$

where the positive constant $\delta$ and the monotone increasing function $Q$ are independent of $t$, $\tau$, and $v$.

Proof.  Multiplying (8.6) by $\partial_t v+\delta v$, where $\delta>0$ is sufficiently small, and arguing in the standard way, we obtain an analogue of the identity (7.16), where $z_n=0$ and the non-linearity $f$ is replaced by $f_L(u):=f(u)+Lu$. Since $f$ satisfies (4.2), one can easily verify that for sufficiently large $L$

$$\begin{equation*} F_L(u)\geqslant0,\quad f_L(u)u-F_L(u)\geqslant0, \end{equation*}$$

and (7.16) implies that

$$\begin{equation*} \frac{d}{dt}\mathscr E(\xi_v)+\delta\mathscr E(\xi_v)\leqslant0. \end{equation*}$$

Applying the Gronwall inequality and using the fact that

$$\begin{equation*} \frac{1}{4}\|\xi_v\|^2_{\mathscr{E}}\leqslant\mathscr E(\xi_v)\leqslant Q(\|\xi_v\|_{\mathscr{E}}), \end{equation*}$$

we end up with the desired estimate for the energy norm of $v$. To get control of the Strichartz norm, we apply the energy-to-Strichartz estimate (5.5) to (8.6) and get that

$$\begin{equation*} \|v\|_{L^4(t,t+1;L^{12})}\leqslant Q(\|\xi_v(t)\|_{\mathscr{E}})\leqslant Q(\|\xi_v(\tau)\|_{\mathscr{E}}). \end{equation*}$$

Next we again use the fact that $f(0)=0$, which together with the fact that $f$ has at most quintic growth rate gives us the control

$$\begin{equation*} \|f_L(u)\|_{L^1(t,t+1;H)}\leqslant C(1+\|u\|_{L^4(t,t+1;L^{12})}^4) \|\xi_v\|_{L^\infty(t,t+1;\mathscr{E})}\leqslant Q(\|\xi_u(\tau)\|_{\mathscr{E}})e^{-\delta(t-\tau)}. \end{equation*}$$

Finally, treating the term $f_L(u)$ as an external force and applying the Strichartz estimate to the linear equation obtained, we arrive at the desired decaying Strichartz estimate for $v$ and thereby finish the proof of the lemma. $\square$

We are now ready to treat the most complicated $w$-component of the solution $u$. We do this in two steps: in the first step we get an estimate growing exponentially in time, which will be refined in the second step.

Lemma 8.5.  Let the above assumptions hold. Then the solution $w(t)$ satisfies the estimate

$$\begin{equation} \|\xi_w(t)\|_{\mathscr{E}^\alpha}+\|w\|_{L^4(\tau,t;H^{\alpha,12})}\leqslant \bigl(Q(\|\xi_u(\tau)\|_{\mathscr{E}})+Q(\|\mu\|_W)\bigr)e^{K(t-\tau)}, \end{equation} \tag{ 8.10 }$$

where $\alpha\in(0,2/5)$ is a sufficiently small positive exponent, and the monotone increasing functions $K=K(\|\xi_u(\tau)\|_\mathscr{E}+\|\mu\|_W)$ and $Q$ are independent of $t$ and $\tau$, and of the concrete choice of $u$ and $\mu$.

Proof.  Treating the non-linearity in (8.7) as an external force and applying the $\mathscr{E}^\alpha$-energy and Strichartz estimates to this linear equation, we get that

$$\begin{equation} \begin{aligned} \, \nonumber &\|\xi_w(t)\|_{\mathscr{E}^\alpha}+ \biggl(\int_\tau^te^{-4\delta(t-s)}\|w(s)\|^4_{H^{\alpha,12}}\,ds\biggr)^{1/4} \\ &\qquad\leqslant C\int_\tau^te^{-\delta(t-s)} (\|f(\theta+v+w)-f(v)\|_{H^\alpha}+L\|v(s)\|_{H^1})\,ds, \end{aligned} \end{equation} \tag{ 8.11 }$$

where $\delta>0$ is a sufficiently small number (see (2.5)). To estimate the non-linear term we use the key inequality (10.11), which gives

$$\begin{equation} \begin{aligned} \, \nonumber &\|f(\theta+v+w)-f(v)\|_{H^\alpha}\leqslant C(1+\|\theta+w\|_{L^{12}}+\|v\|_{L^{12}})^{4-\alpha} \\ &\qquad\times(1+\|\theta+w\|_{H^1}+\|v\|_{H^1})^{\alpha} \|\theta+w\|_{H^{1+\alpha}}^{1-\alpha}\|\theta+w\|^\alpha_{H^{\alpha,12}} \end{aligned} \end{equation} \tag{ 8.12 }$$

and

$$\begin{equation*} \begin{aligned} \, \|\theta+w\|_{H^{1+\alpha}}^{1-\alpha}\|\theta+w\|^\alpha_{H^{\alpha,12}} &\leqslant \|\theta\|_{H^{1+\alpha}}^{1-\alpha} \|\theta\|^\alpha_{H^{\alpha,12}}+ \|w\|_{H^{1+\alpha}}^{1-\alpha}\|w\|^\alpha_{H^{\alpha,12}} \\ &\qquad+\|\theta\|_{H^{1+\alpha}}^{1-\alpha}\|w\|^\alpha_{H^{\alpha,12}}+ \|w\|_{H^{1+\alpha}}^{1-\alpha}\|\theta\|^\alpha_{H^{\alpha,12}}. \end{aligned} \end{equation*}$$

Using the Hölder inequality together with the control (8.8) for the $\theta$-component, we get that

$$\begin{equation} \begin{aligned} \, \nonumber &\int_\tau^te^{-\delta(t-s)}(1+\|\theta+w\|_{L^{12}}+\|v\|_{L^{12}})^{4-\alpha} (1+\|\theta+w\|_{H^1}+\|v\|_{H^1})^{\alpha} \\ \nonumber &\qquad\qquad\times \|w\|_{H^{1+\alpha}}^{1-\alpha}\|w\|^\alpha_{H^{\alpha,12}}\,ds \\ \nonumber &\qquad\leqslant C\biggl(\,\int_\tau^t\exp\biggl\{-\delta(t-\tau) \frac{1-\alpha}{1-\alpha/4}\biggr\}(1+\|\theta+w\|^4_{L^{12}}+ \|v\|^4_{L^{12}}) \\ \nonumber &\qquad\qquad\times(1+\|\theta+w\|_{H^1}+\|v\|_{H^1})^{\alpha/(1-\alpha/4)} \|\xi_w(s)\|_{\mathscr{E}^\alpha}^{(1-\alpha)/(1-\alpha/4)}\, ds\biggr)^{1-\alpha/4} \\ \nonumber &\qquad\qquad\times\biggl(\,\int_\tau^te^{-4\delta(t-s)} \|w(s)\|^4_{H^{\alpha,12}}\,ds\biggr)^{\alpha/4} \\ \nonumber &\qquad\leqslant\frac{1}{8}\biggl(\,\int_\tau^te^{-4\delta(t-s)} \|w(s)\|^4_{H^{\alpha,12}}\,ds\biggr)^{1/4} \\ &\qquad\qquad+C\biggl(\,\int_\tau^te^{-\delta'(t-\tau)}l_{\theta,v,w}(s) \|\xi_w(s)\|_{\mathscr{E}^\alpha}^{(1-\alpha)/(1-\alpha/4)}\, ds\biggr)^{(1-\alpha/4)/(1-\alpha)}, \end{aligned} \end{equation} \tag{ 8.13 }$$

where $\delta'=\delta(1-\alpha)/(1-\alpha/4)$ and

$$\begin{equation*} \begin{aligned} \, l_{\theta,v,w}(s)&:=(1+\|\theta(s)+w(s)\|^4_{L^{12}} +\|v(s)\|^4_{L^{12}}) \\ &\qquad\times (1+\|\theta(s)+w(s)\|_{H^1}+\|v(s)\|_{H^1})^{\alpha/(1-\alpha/4)}. \end{aligned} \end{equation*}$$

Estimation of the three other terms in (8.12) containing the $H^{1+\alpha}$- and $H^{\alpha,12}$-norms of $\theta$ is analogous but even simpler due to the control (8.8). According to the estimates already obtained, we have

$$\begin{equation} \int_{t}^{t+1}l_{\theta,v,w}(s)\,ds\leqslant Q_1= Q_1(\|\xi_u(\tau)\|_{\mathscr{E}}+\|\mu\|_{W}),\qquad t\geqslant\tau, \end{equation} \tag{ 8.14 }$$

and substituting (8.13) into the right-hand side of (8.11), we arrive at the inequality

$$\begin{equation} \begin{aligned} \, \nonumber &\|\xi_w(t)\|_{\mathscr{E}^\alpha}+\biggl(\,\int_\tau^te^{-4\delta(t-s)} \|w(s)\|^4_{H^{\alpha,12}}\,ds\biggr)^{1/4} \\ &\qquad\leqslant C\biggl(\,\int_\tau^te^{-\delta'(t-\tau)}l_{\theta,v,w}(s) \|\xi_w(s)\|_{\mathscr{E}^\alpha}^{(1-\alpha)/(1-\alpha/4)}\, ds\biggr)^{(1-\alpha/4)/(1-\alpha)}+Q_2, \end{aligned} \end{equation} \tag{ 8.15 }$$

where the constant $Q_2$ depends only on $\|\xi_u(\tau)\|_{\mathscr{E}}$ and $\|\mu\|_W$. Introducing the function

$$\begin{equation*} Y(t):=\|\xi_w(t)\|_{\mathscr{E}^\alpha}^{(1-\alpha)/(1-\alpha/4)} \end{equation*}$$

and raising both sides of (8.15) to the power $(1-\alpha)/(1-\alpha/4)$, we get finally that

$$\begin{equation} Y(t)\leqslant Q+C\int_\tau^te^{-\delta'(t-s)}l_{\theta,v,w}(s)Y(s)\,ds \end{equation} \tag{ 8.16 }$$

for some new constant $Q$ depending on $\xi_u(\tau)$ and $\|\mu\|_W$. The Gronwall inequality applied to this estimate, together with (8.14), gives the desired estimate (8.10) and finishes the proof of the lemma. $\square$

We now state (following [43]) a corollary of the estimates obtained that is crucial for what follows.

Corollary 8.6.  Let the above assumptions hold and let $\xi_u(\tau)\in\mathscr B$, where $\mathscr B$ is a uniformly absorbing set for the equation (1.1). Then for any $\varepsilon>0$ there exists a splitting $w(t)=\overline w(t)+\widetilde w(t)$ of the solution $w(t)$ of (8.7) such that

$$\begin{equation} \int_s^t\|\widetilde w(\kappa)\|^4_{L^{12}}\,d\kappa+ \int_s^t\|\xi_{\widetilde w}(\kappa)\|_{\mathscr{E}}^{\alpha/(1-\alpha/4)}\, d\kappa\leqslant C_\varepsilon+\varepsilon(t-s),\quad t\geqslant s\geqslant\tau, \end{equation} \tag{ 8.17 }$$

and

$$\begin{equation} \|\xi_{\overline w}(t)\|_{\mathscr{E}^\alpha}+ \|\overline w\|_{L^4(t,t+1;H^{\alpha,12})}\leqslant C_\varepsilon,\quad t\geqslant\tau, \end{equation} \tag{ 8.18 }$$

where the constant $C_\varepsilon$ depends only on $\varepsilon$ (and is independent of $t$, $s$, $\tau$, and $\xi_u(\tau)\in \mathscr B$). Moreover,

$$\begin{equation} \|\xi_{\overline w}(t)\|_{\mathscr{E}}+\|\overline w\|_{L^4(t,t+1;L^{12})}+ \|\xi_{\widetilde w}(t)\|_{\mathscr{E}}+ \|\widetilde w\|_{L^4(t,t+1;L^{12})}\leqslant C,\quad t\geqslant\tau, \end{equation} \tag{ 8.19 }$$

where $C$ is independent also of $\varepsilon$.

Proof.  Note that by the estimates (8.9) and (8.8) it is sufficient to construct the desired splitting $u(t)=\overline u(t)+\widetilde u(t)$ for $u$ only. To do this we fix a large $T=T(\varepsilon)$ (actually $T\sim 1/\varepsilon$) and construct the splitting (8.4) at the points $\tau_0=\tau$, $\tau_1=\tau+T$, $\tau_2=\tau+2T$, and so on. Namely, let $\theta_n(t)$, $v_n(t)$, and $w_n(t)$ denote the solutions of the problems (8.5), (8.6), and (8.7), respectively, where the initial moment of time $\tau$ is replaced by $\tau+nT$, and define

$$\begin{equation} \widetilde u(t):=v_n(t)\quad\text{and}\quad \overline u(t):=\theta_n(t)+w_n(t),\qquad t\in[\tau+nT,\tau+(n+1)T). \end{equation} \tag{ 8.20 }$$

Then as elementary calculations based on (8.9) show, the function $\widetilde u(t)$ satisfies (8.17) and (8.19). In turn, the estimates (8.8) and (8.10), together with the dissipative estimate for the solution $u(t)$, guarantee that the function $\overline u$ satisfies (8.18) and (8.19). Finally, to obtain the desired splitting of $w$, we just need to take

$$\begin{equation*} \widetilde w(t)=\widetilde u(t)-v(t)\quad\text{and}\quad \overline w(t)=\overline u(t)-\theta(t), \end{equation*}$$

and the corollary is proved. $\square$

We are now ready to refine Lemma 8.5.

Lemma 8.7.  Let the above assumptions hold and let $\xi_u(\tau)\in\mathscr B$. Then the solution $w$ of the problem (8.7) satisfies the estimate

$$\begin{equation} \|\xi_{w}(t)\|_{\mathscr{E}^\alpha}+ \|w\|_{L^4(t,t+1;H^{\alpha,12})}\leqslant C, \end{equation} \tag{ 8.21 }$$

where the constant $C$ is independent of $t$, $\tau$, and $\xi_u(\tau)\in\mathscr B$.

Proof.  We refine the estimates (8.12) and (8.13) using the result of Corollary 8.6. To this end, we first note that we may assume without loss of generality that $f'(0)=0$. Indeed, the extra term $|f'(0)|\|w(t)+\theta(t)\|_{H^\alpha}$ is easily controllable by the estimates obtained above in the energy space $\mathscr{E}$. Next, we write the difference $f(\theta+v+w)-f(v)$ as follows

$$\begin{equation} \begin{aligned} \, \nonumber |f(\theta+v+w)-f(v)|&\leqslant |f(\theta+v+w)-f(v+\widetilde w)|+ |f(v+\widetilde w)-f(v)| \\ \nonumber &\leqslant |f(\theta+v+w)-f(v+\widetilde w)|+ \biggl|\int_0^1f'(v+\kappa\widetilde w)\,d\kappa\,w\biggr| \\ &\qquad+\biggl|\int_0^1f'(v+\kappa\widetilde w)\,d\kappa\,\overline w\biggr|. \end{aligned} \end{equation} \tag{ 8.22 }$$

The first term on the right-hand side of (8.22) is controlled exactly as in (8.12):

$$\begin{equation} \begin{aligned} \, \nonumber &\|f(\theta+v+w)-f(v+\widetilde w)\|_{H^\alpha}\leqslant C(1+\|\theta+\overline w\|_{L^{12}}+\|v+\widetilde w\|_{L^{12}})^{4-\alpha} \\ \nonumber &\qquad\qquad\times(1+\|\theta+\overline w\|_{H^1}+ \|v+\widetilde w\|_{H^1})^{\alpha} \|\theta+\overline w\|_{H^{1+\alpha}}^{1-\alpha}\|\theta+ \overline w\|_{H^{\alpha,12}}^{\alpha} \\ &\qquad\leqslant C_\varepsilon(1+\|\theta\|_{H^{\alpha,12}}^4+ \|v\|_{L^{12}}^4+\|\overline w\|_{H^{\alpha,12}}^4+ \|\widetilde w\|_{L^{12}}^4). \end{aligned} \end{equation} \tag{ 8.23 }$$

The third term is estimated similarly, using (10.4):

$$\begin{equation} \begin{aligned} \, \nonumber &\biggl\|\int_0^1f'(v+\kappa\widetilde w)\overline w\, d\kappa\biggr\|_{H^\alpha}\leqslant C(1+\|v+\widetilde w\|_{L^{12}}+\|v\|_{L^{12}})^{4-\alpha} \\ \nonumber &\qquad\qquad\times(1+\|v+\widetilde w\|_{H^1}+\|v\|_{H^1})^{\alpha} \|\overline w\|_{H^{1+\alpha}}^{1-\alpha} \|\overline w\|_{H^{\alpha,12}}^{\alpha} \\ &\qquad\leqslant C_\varepsilon(1+\|v\|_{L^{12}}^4+ \|\widetilde w\|_{L^{12}}^4+\|\overline w\|_{H^{\alpha,12}}^4). \end{aligned} \end{equation} \tag{ 8.24 }$$

Thus, in view of the estimates (8.18), (8.8), (8.9), and (8.19) we have

$$\begin{equation} \begin{aligned} \, \nonumber &\int_\tau^t e^{-\delta(t-s)}\|f(\theta(s)+v(s)+w(s))-f(v(s)+ \widetilde w(s))\|_{H^\alpha}\,ds \\ &\qquad+\int_\tau^t e^{-\delta(t-s)}\biggl\|\int_0^1f'(v(s)+ \kappa\widetilde w(s))\overline w(s)\,d\kappa\biggr\|_{H^\alpha}\,ds \leqslant C_\varepsilon. \end{aligned} \end{equation} \tag{ 8.25 }$$

So we only need to estimate the second term on the right-hand side of (8.22). To this end we use the assumption that $f'(0)=0$ and apply (10.10) to get that

$$\begin{equation*} \begin{aligned} \, &\biggl\|\int_0^1f'(v(s)+\kappa\widetilde w(s))w(s)\, d\kappa\biggr\|_{H^\alpha} \\ &\qquad\leqslant C(1+\|v\|_{L^{12}}+ \|\widetilde w\|_{L^{12}})^{4-\alpha}(\|v\|_{H^1}+ \|\widetilde w\|_{H^1})^{\alpha}\|w\|_{\mathscr{E}^\alpha}^{1-\alpha} \|w\|_{H^{\alpha,12}}^\alpha. \end{aligned} \end{equation*}$$

Arguing as in (8.12), we then get that

$$\begin{equation} \begin{aligned} \, \nonumber &\int_\tau^t e^{-\delta(t-s)}\biggl\|\int_0^1f'(v(s)+ \kappa\widetilde w(s))\overline w(s)\,d\kappa\biggr\|_{H^\alpha}\,ds \\ \nonumber &\qquad\leqslant \frac{1}{2}\biggl(\,\int_\tau^te^{4\delta(t-s)} \|w(s)\|^4_{H^{\alpha,12}}\,ds\biggr)^{1/4} \\ &\qquad\qquad+\biggl(\,\int_\tau^t e^{-\delta'(t-s)}l(s) \|\xi_w(s)\|_{\mathscr{E}^\alpha}^{(1-\alpha)/(1-\alpha/4)}\, ds\biggr)^{(1-\alpha/4)/(1-\alpha)}, \end{aligned} \end{equation} \tag{ 8.26 }$$

where

$$\begin{equation} \begin{aligned} \, \nonumber l(s)&:=C(1+\|v(s)\|_{L^{12}}+\|\widetilde w(s)\|_{L^{12}})^4(\|v(s)\|_{H^1}+ \|\widetilde w(s)\|_{H^1})^{\alpha/(1-\alpha/4)} \\ &\kern2pt\leqslant C(\|v(s)\|_{L^{12}}^4+\|\widetilde w(s)\|^4_{L^{12}}+ \|\xi_{\widetilde w}(s)\|_{\mathscr{E}}^{\alpha/(1-\alpha/4)}+ \|\xi_{v}(s)\|_{\mathscr{E}}^{\alpha/(1-\alpha/4)}). \end{aligned} \end{equation} \tag{ 8.27 }$$

It is important that the constant $C$ here is independent of $\varepsilon$. Therefore, due to (8.17) and (8.9) we have

$$\begin{equation} \int_s^tl(\kappa)\,d\kappa\leqslant C_\varepsilon+\varepsilon(t-s),\qquad t\geqslant s\geqslant\tau, \end{equation} \tag{ 8.28 }$$

and substituting the resulting estimates into (8.11), we finally get that the function $Y(t)=\|\xi_w(t)\|_{\mathscr{E}^{\alpha}}^{(1-\alpha)/(1-\alpha/4)}$ satisfies a refined analogue of (8.16):

$$\begin{equation} Y(t)+\biggl(\,\int_\tau^te^{-4\delta(t-s)} \|w(s)\|^4_{H^{\alpha,12}}\,ds\biggr)^{1/4}\leqslant C_\varepsilon+\int_\tau^te^{-\delta'(t-s)}l(s)Y(s)\,ds. \end{equation} \tag{ 8.29 }$$

To derive the desired estimate (8.21) from (8.29), we need the following version of the Gronwall lemma.

Lemma 8.8.  Let the function $Y\in C_{\mathrm{loc}}([\tau,\infty))$ satisfy the estimate

$$\begin{equation*} Y(t)\leqslant C+\int_\tau^te^{-\delta'(t-s)}l(s)Y(s)\,ds,\qquad t\geqslant\tau, \end{equation*}$$

for some constants $C$ and $\delta$ and a non-negative function $l\in L^1_{\mathrm{loc}}([\tau,\infty))$. Then

$$\begin{equation} Y(t)\leqslant C\biggl(1+\int_\tau^t \exp\biggl\{-\delta'(t-s)+ \int_s^tl(\kappa)\,d\kappa\biggr\}l(s)\,ds\biggr). \end{equation} \tag{ 8.30 }$$

The proof of this lemma follows almost word for word the proof of the usual Gronwall lemma and is thus omitted.

Applying the estimate (8.30) to (8.29) and using (8.28) (with the parameter $\varepsilon$ fixed such that $\varepsilon<\delta'$), we derive the desired estimate (8.21) and thereby finish the proof of Lemma 8.7. $\square$

Now we are ready to complete the proof of the main theorem.

Proof of Theorem 8.1. According to the estimates (8.8), (8.9), and (8.21), the set

$$\begin{equation*} \mathscr B_\alpha:=\{\xi\in\mathscr{E}^\alpha\colon \|\xi\|_{\mathscr{E}^\alpha}\leqslant R\} \end{equation*}$$

is a compact uniformly attracting set for the process $U_\mu(t,\tau)$ in $\mathscr{E}$ if $R$ is large enough. Thus, the process $U_\mu(t,\tau)$ is uniformly asymptotically compact and possesses a uniform attractor $\mathscr A_{\mathrm{un}}$ in the strong topology of $\mathscr{E}$. Moreover, $\mathscr A\subset \mathscr B_\alpha$. This completes the proof of Theorem 8.1. $\square$

The next corollary gives the global well-posedness and dissipativity of the process $U_\mu(t,\tau)$ in the higher energy space $\mathscr{E}^\alpha$.

Corollary 8.9.  Let the assumptions of Theorem 8.1 hold, and in addition let $\xi_u(\tau)\in\mathscr E^\alpha$. Then the solution $u$ of (1.1) satisfies $\xi_u(t)\in\mathscr{E}^\alpha$ for all $t\geqslant\tau$ and the following estimate holds:

$$\begin{equation} \|\xi_u(t)\|_{\mathscr{E}^\alpha}+\|u\|_{L^4(t,t+1;H^{\alpha,12})}\leqslant Q(\|\xi_u(\tau)\|_{\mathscr{E}^\alpha})e^{-\delta(t-\tau)}+Q(\|\mu\|_{W}), \end{equation} \tag{ 8.31 }$$

where the constant $\delta>0$ and the monotone increasing function $Q$ are independent of $t$, $\tau$, $\mu$, and $u$.

Indeed, the proof of this estimate is based on the result of Corollary 8.6 and can be obtained as in the derivation of (8.21) (and even more simply, since we may take $v(t)=0$ and pose the non-zero initial conditions directly for the $w$-component). For this reason we leave the detailed proof of this corollary to the reader.

Remark 8.10.  Note that Theorem 8.1 and Corollary 8.9 are formally proved only under the assumption that $\alpha<2/5$, and we do not know how to obtain more regularity of the attractor $\mathscr A_{\mathrm{un}}$ in one step even in the case where $\mu$ and $f$ are smooth. For instance, it would be interesting to get $\mathscr{E}^1$-regularity without using fractional Sobolev spaces. The problem is related to the restriction on the exponent $\alpha$ in the key lemma, Lemma 10.2. However, higher regularity can be easily obtained in several steps using standard bootstrap arguments. Moreover, the most difficult step is exactly the first: to obtain the $\mathscr{E}^\alpha$ regularity of solutions for small positive $\alpha$. Since the non-linearity is no longer critical in $\mathscr{E}^\alpha$, for further iterations one can use a standard linear decomposition to further improve the regularity. Namely, in (8.4) we may take $v=0$ and $\xi_{\theta}\big|_{t=\tau}=\xi_u\big|_{t=\tau}$. For instance, if $\alpha>1/8$, we have

$$\begin{equation*} \|f(u)\|_{L^1(t,t+1;H^1)}\leqslant C(1+\|u\|_{L^4(t,t+1;H^{\alpha,12})}^4) \|\xi_u\|_{L^\infty(t,t+1;\mathscr{E}^\alpha)}, \end{equation*}$$

and therefore we need only one extra step of iterations to get the $\mathscr{E}^1$-regularity of the attractor.

In this appendix we recall a number of more or less standard results concerning functions of bounded variation ($\mathrm{BV}$-functions) with values in Banach spaces and the associated measures which are used throughout the paper. We restrict ourselves to the case where these functions take values in a separable Hilbert space $H$ (see [7], [8], [13], [28], and the references therein for more details).

Definition 9.1.  A function $\Phi\colon [a,b]\to H$ is a function of bounded variation (a $\mathrm{BV}$-function) if

$$\begin{equation} \operatorname{Var}_a^b(\Phi,H):=\sup\biggl\{\,\sum_{j=1}^N\|\Phi(t_j)- \Phi(t_{j-1})\|_H\biggr\}<\infty, \end{equation} \tag{ 9.1 }$$

where the supremum is taken over all finite $N$ and all partitions $a=t_0<t_1<\cdots<t_N=b$ of the interval $[a,b]$.

We also recall the elementary properties of the variation introduced:

• 1)  
  $\operatorname{Var}_a^b(\Phi,H)=\operatorname{Var}_a^c(\Phi,H)+\operatorname{Var}_c^b(\Phi,H)$ if $a<c<b$;
• 2)  
  $\operatorname{Var}_a^b(\Phi_1+\Phi_2,H) \leqslant\operatorname{Var}_a^b(\Phi_1,H)+\operatorname{Var}_a^b(\Phi_2,H)$;
• 3)  
  $\operatorname{Var}_a^b(\alpha\Phi,H)=|\alpha|\operatorname{Var}_a^b(\Phi,H)$;
• 4)  
  the function $\varphi(t):=\operatorname{Var}_a^t(\Phi,H)$ is monotone increasing and $\operatorname{Var}_x^y(\Phi,H) =\varphi(y)-\varphi(x)$;
• 5)  
  the function $\Phi(t)$ is continuous (left/right-continuous) at $t=t_0$ if and only if the same is true for $\varphi(t)$.

Note that if $\Phi$ is a $\mathrm{BV}$-function, then $\varphi$ is a scalar non-decreasing and bounded function, so it is continuous away from an at most countable set of points (indeed, for every $n$ the number of jumps of $\varphi$ which are larger than $1/n$ must be finite). Therefore, property 5) guarantees that $\Phi(t)$ also has an at most countable number of discontinuities. Furthermore, due to monotonicity, the right/left limits $\varphi(t+0)$ and $\varphi(t-0)$ exist for all $t\in[a,b]$, and therefore simple arguments show that the right/left limits of $\Phi$ also exist at every point in $[a,b]$.

Definition 9.2.  We denote by $V_0(a,b;H)$ the Banach space of all $\mathrm{BV}$-functions $\Phi$ on $[a,b]$ with values in $H$ such that $\Phi(a)=0$ and $\Phi(t)$ is left-continuous at every interior point of $[a,b]$ (that is, $\Phi$ may have a jump at $b$). The norm in this space is given by

$$\begin{equation*} \|\Phi\|_{V_0}:=\operatorname{Var}_a^b(\Phi,H). \end{equation*}$$

As usual, every $\Phi\in V_0$ defines a vector-valued measure $\mu_\Phi$ on the semiring generated by the subintervals of $[a,b]$ via the formulae

$$\begin{equation} \begin{alignedat}{2} \mu_\Phi((s,t])&:=\Phi(t+0)-\Phi(s+0),&\qquad \mu_\Phi([s,t])&:=\Phi(t+0)-\Phi(s-0), \\ \mu_\Phi((s,t))&:=\Phi(t-0)-\Phi(s+0),&\qquad \mu_\Phi([s,t))&:=\Phi(t-0)-\Phi(s-0), \end{alignedat} \end{equation} \tag{ 9.2 }$$

where $a\leqslant s\leqslant t\leqslant b$, and we use the notation $(t,t]=[t,t)=(t,t)=\varnothing$, $[t,t]={t}$ and $\Phi(b+0):=\Phi(b)$, $\Phi(a-0):=0$. As usual, the assumption of left-continuity implies the $\sigma$-additivity of $\mu_\Phi$ on the algebra generated by subintervals. Moreover, this measure can be extended in a unique way to the $\sigma$-algebra of Borel subsets of $[a,b]$ and gives a $\sigma$-additive Borel vector measure $\mu_\Phi$ of finite total variation $|\mu_\Phi|([a,b])<\infty$. We recall that for any Borel set $A\subset [a,b]$ the total variation is

$$\begin{equation} |\mu_\Phi|(A):=\sup\biggl\{\,\sum_{n=1}^\infty\|\mu_\Phi(A_n)\|_{H}\biggr\}, \end{equation} \tag{ 9.3 }$$

where the supremum is taken over all countable disjoint Borel partitions of $A$. It is also known that $|\mu_\Phi|$ is a scalar positive $\sigma$-additive measure generated by the distribution function $\varphi(t):=\operatorname{Var}_a^t(\Phi,H)$, that is, the formulae (9.2) remain valid if one replaces $\mu_\Phi$ by $|\mu_\Phi|$ on the left and $\Phi$ by $\varphi$ on the right. In particular, we have

$$\begin{equation} |\mu_\Phi|([a,b])=\varphi(b+0)-\varphi(a-0)=\operatorname{Var}_a^b(\Phi,H). \end{equation} \tag{ 9.4 }$$

Conversely, with any $\sigma$-additive $H$-valued measure $\mu$ defined on the Borel $\sigma$-algebra of $[a,b]$ with finite total variation $|\mu|([a,b])<\infty$ we can associate a $\mathrm{BV}$-function $\Phi_\mu\colon [a,b]\to H$ in $V_0([a,b];H)$ by the formula