Necessary and sufficient conditions for the existence of a classical solution of the mixed problem for the wave equation on a graph

We study a mixed problem for the wave equation with integrable potential on the simplest geometric graph consisting of two ring edges that touch at a point. We use a new approach in the Fourier method to obtain necessary and sufficient conditions for the existence of a classical solution. We do not use refined asymptotic formulas for the eigenvalues and any information on the eigenfunctions. The solution is represented by a rapidly convergent series.


Introduction
We consider the simplest geometric graph consisting of two ring edges that touch at a point (at the node of the graph). Parametrizing each edge by the interval [0, 1], we study the following mixed problem for the wave equation on this graph: x ∈ [0, 1], t ∈ (−∞, +∞), (j = 1, 2), We assume that q j (x) ∈ L[0, 1] are complex-valued. Conditions (2), (3) are generated by the structure of the graph [1,2]. Research some problems on geometric graphs can also be found in [3]- [8].
In this problem the application of the Fourier method causes difficulties associated with the fact that the eigenvalues of the corresponding spectral problem might be multiple. These difficulties can be coped with by applying the resolvent approach [9]. By the resolvent approach a sufficient conditions for the existence of classical solution of a mixed problem on such a graph is obtained in [10]- [11]. Note that we did not use refined asymptotic formulas for the eigenvalues and any information on the eigenfunctions. We used Krylovs ideas [12, Chapter VI] concerning the convergence acceleration of Fourier-like series (see also [13]- [15]. 2 Sufficient conditions on the function ϕ j (x) for the existence of a classical Fourier method solution of problem (1)-(4) (Eq. (1) is satisfied almost everywhere) were obtained in [11]. These conditions are as follows: T denotes the transpose. These conditions, except for condition Lϕ ∈ L 2 2 [0, 1], are necessary for the existence of a classical solution of the problem. Here we remove this additional condition and thus obtain the necessary and sufficient conditions for the existence of the classical solution.
Here we will use a different approach proposed by A. P. Khromov in [16,17], again based on the ideas of accelerating the convergence of series, but implying a different transformation of the formal series. This approach allows us to obtain the necessary and sufficient conditions for the existence of classical solutions. In this case, the solution is represented as a series that converges at an exponential rate.

Uniqueness of a classical solution
Here we consider the following problem, which is more general than (1)-(4): x ∈ [0, 1], t ∈ (−∞, +∞), (j = 1, 2), We assume all functions in (1) . Further we will also use the notation: A classical solution is defined as a function u(x, t) = (u 1 (x, t), u 2 (x, t)) T such that u j (x, t) and their first derivatives with respect to x and t are absolutely continuous, and satisfies the boundary and initial conditions (6)-(8) and the differential equation (5) almost everywhere. Therefore, we assume that the vector functions ϕ(x), ψ(x) and ϕ (x) are absolutely continuous and such that satisfy the following conditions:

the uniqueness condition), then this solution is unique and can be represented by the formula
where the series on the right-hand side converges absolutely and uniformly with respect to x ∈ [0, 1] for each t ≥ 0. Here R λ = (L − λE) −1 is the resolvent of the operator (λ is the spectral parameter, and E is the identity operator); the notation R λ (f (·, τ)) indicates that the operator R λ is applied to f (x, τ ) with respect to the variable x; λ = ρ 2 , Re ρ ≥ 0, γ n is the image of the circleγ n = {ρ||ρ − nπ| = δ} in the λ-plane, δ > 0 is sufficiently small, r is sufficiently large and fixed; and n 0 is a number such that for each n ≥ n 0 the contour γ n lies outside the circle |λ| = r and all eigenvalues of L are inside γ n . The proof of Theorem 1 is analogous to the proof of Theorem 1 in [16] (see also in [18,19]).

Transformation of the formal solution
Let's go back to problem (1)-(4). The Fourier method is related to the spectral problem Ly = λy for operator L.
The formal solution u(x, t) = (u 1 (x, t), u 2 (x, t)) T of problem (1)-(4) produced by the Fourier method can be represented as where r > 0 is fixed and such that all the eigenvalues λ n , with n < n 0 , belong to the disk |λ| < r, and there are no eigenvalues of L on the contour |λ| = r; γ n are the contours of sufficiently small radius in λ-plane such that all the eigenvalues of operator L and L 0 with n ≥ n 0 are only inside γ n (see [9,18]). The formal solution can be represented as where U 0 (x, t) is (10) with R 0 λ instead of R λ , and U 1 (x, t) is the series (10) in which R λ is replaced with R λ − R 0 λ . The series U 0 (x, t) is solution of the problem (1)-(4) with q j (x) = 0 and it converges absolutely and uniformly over all x and t [11]. Let's denote its sum A 0 (x, t). According to [11] follows the statement

Study of U 1 (x, t).
In [9]- [11] the series U 1 (x, t) was investigated using resolvent estimates. Now, as in [16], we consider it as a solution to the problem First, consider the problem (12)- (15) with an arbitrary right-hand side f (x, t) and Q(x) = 0.

Theorem 3
If the function f (x, t) is continuously differentiable with respect to x and t and t), then the series for the formal solution of problem converges for all x and t, and its sum can be represented in the form where the functionF (η, τ ) = f (η, τ ) for η ∈ [0, 1] andF (η, τ ) continues on the entire axis using the relations (11).
Next, we represent U 1 (x, t) as: where now the function A 1 (x, t) has the form (16) Just as in [16,Theorem 3], one can prove

Theorem 4 If u(x, t) is a classical solution of problem (1)-(4) with the uniqueness condition, then the function
, and continuously differentiable with respect to x and t, and its derivative A 1x (x, t) (A 1t (x, t)) is absolutely continuous with respect to x (respectively, t); conditions (13)- (15) are satisfied for A 1 (x, t); and almost everywhere with respect to (x, t) ∈ (−∞, +∞) × [0, ∞) one has the relation whereQ(x) continues on the entire axis using the relations (11). The equation being true for all x and t such that the functions

The classical solution
We will study the function U 2 (x, t) by analogy with the function U 1 (x, t) above, i.e., based on the fact that U 2 (x, t) is a classical solution of problem (12)- (15) with the condition ∂ 2 U 2 (x,t) As a result, we obtain where the same formula holds for the function A 2 (x, t) as for the function A 1 (x, t) above with F 0 (η, τ ) replaced with F 1 (η, τ ) = −Q(η)A 1 (η, τ ) and U 3 (x, t) is the classical solution of problem (12)-(15) with U 2 (x, t) and F 2 (x, t) replaced with U 3 (x, t) and F 2 (x, t), respectively. Continuing this process ad infinitum, we obtain where the same formula takes place for the function A n (x, t) as for the function is the solution of the problem obtained from (12)-(15) by the replacement of U 1 (x, t) and F 0 (x, t) with U n+1 (x, t) and F n (x, t) = −Q(x)A n (x, t), respectively. By induction, for A n (x, t) we obtain the theorem similar to Theorem 4. Thus, the representation is true: Using Theorem 3 and [16, Lemma 10] we can prove Lemma 1 Let T be an arbitrary positive number, and let N be the least positive integer such that T ≤ N . Then  and F n (x, t) = −Q(x)A n (x, t) for x ∈ [0, 1], and continues on the entire axis using the relations (11). Here the series converges absolutely and uniformly with respect to x, t ∈ Q T for each T > 0. Proof. As follows from Lemma 1, the series A(x, t) converges absolutely and uniformly with respect to x, t ∈ Q T for each T > 0. Further, for ρ ∈γ n we have the estimates

Theorem 5 If u(x, t) is a classical solution of problem (1)-(4) such that
([ ] k denotes the k-th component of the vector function), the constant C is independent of η, τ and n, and, by Lemma 1, we have the estimates The proof of the theorem is complete.