Generalized solvability of a problem with a dynamic boundary condition for the hyperbolic equation

In this article, we consider a problem with a dynamic boundary condition for a one-dimensional hyperbolic equation. The conditions on coefficients and a source term providing a unique solvability of a problem are derived.


Introduction
Vibration problems are of great importance in engineering and nowadays have been studied by many researchers.
Construction structures and buildings are highly susceptible to both natural and man-made dynamic impacts. For example, wind and seismic impacts, loads from equipment, moving transport, pedestrians.
The energy of the oscillations of the engineering systems is gradually dissipated due to internal friction in the material and external resistance. This fact affects the vibrational process, and the decrease in the intensity of external dynamic influences leads to a fading of fluctuations. Researchers carry out dynamic calculations of structures, identify the dynamic characteristics (frequency, form of own vibrations and so on). It should be taken take into account the effect of internal damping, which extinguishes fluctuations due to friction in the material and thus affects the overall vibrational process. It is rather known how to take into account the effects of external friction (external oscillation), whereas a problem to account into consideration internal friction is often more difficult.
Turning to mathematical terms, we get a problem with nonlocal conditions, which describe the model of internal friction (nonlocal damping of the material).
Nowadays various nonlocal problems for partial differential equations are actively studied. We focus our attention on nonlocal problems with integral conditions for hyperbolic equations. Systematic studies of nonlocal problems with integral conditions originated with the papers by Cannon. These and further investigations of nonlocal problems show that classical methods most widely used to prove solvability of initial-boundary problems break down when applied to nonlocal problems. Nowadays several methods have been devised for overcoming the difficulties arising because of nonlocal conditions.It appears that conditions for the existence and uniqueness of a solution to the nonlocal problem are closely related to the notion of regular boundary conditions. It is known that the system of root functions of an ordinary differential operator with strongly regular boundary conditions form a Riesz basis in L 2 (0, 1).This property is particularly useful for obtaining results on solvability of boundary problems. We consider a problem with nonlocal dynamic condition in the form This condition may be considered as perturbed dynamic condition and we begin our study starting with the case α = 0.
We will introduce a definition of a generalized solution of the problem studied and prove unique solvability of the problem.

Setting of a problem
In interval Q T = (0, l) × (0, T ) consider the equation and set a problem: find a solution to the equation (1), satisfying initial data and boundary conditions As we know [1][2][3][4][5][6][7], solving a problem with dynamic conditions (3) is difficult even for the equation of string vibrations. We consider an equation with arbitrary coefficients depending on both x, t,. It makes it pointless trying to get a solution to a problem by separating variables or taking advantage of a common solution to the equation. However, we managed to prove the unambiguous resolution of the problem in Sobolev's space, as demonstrated in the article. Let us assume that the coefficients of the equation (1) and its right part meet the following conditions: Let's introduce a definition of a generalized solution. To do this, we consider an equality where v(x, t) is an arbitrary smooth function, such that v(T ) = 0, under assumption that u(x, t) is a solution to the problem. After integrating the left side by parts we get Note that all terms in (4) make sense under lesser requirements on the function of u(

The uniqueness of the solution
Suppose there exist two generalized solutions, u 1 (x, t) and u 2 (x, t), to the problem (1). Then their difference u(x, t) = u 1 (x, t) − u 2 (x, t) satisfies the condition u(x, 0) = 0 and identity Note that v t (x, t) = u(x, t). Integrating by parts the terms of left side in (5), we get Let's put the results in (5) Since a(x, t) > 0 in theQ T , then from (6) it follows an inequality Let's estimate the right side of (7). Since c, a, a t , a tt , b, b t , b tt ∈ C(Q T ), there are numbers a 1 , c 0 , b 0 such that |a, a t , a t t| ≤ a 1 , |b, b t , b t t| ≤ b 0 , |c| ≤ c 0 . Then Considering that v(x, 0) = − τ 0 u(x, t)dt and using Cauchy-Bunyakovsky inequality we get Applying the inequality (3.9) [1] we get: To estimate the terms containing the value of v(l, t), apply Cauchy's inequality "with ε " [8]: We'll choose so that µ = a 0 − a 1 > 0 and transfer the integral from right side to left: where M = max{a 1 c( ), a 1 c( )τ }.
We're going to do a number of calculations. Because of Since under the integral in the left part there is a v x (x, 0), and in the right is not, we will introduce a function We will choose τ by the arbitrariness so that µ−2M 1 τ > 0. Let for definiteness µ−2M 1 τ ≥ µ 2 , then for τ where m 0 = min{1, M 2 }. Now from Gronowall's lemma we obtain Continuing this procedure for the τ ∈ [ M 4M 1 ; M 2M 1 ] using algorithm [8], we get zeros again, which follows the singularity of the generalized solution to the problem.

The existence of the solution
Let the functions w k (x) ∈ C 2 (0, l) linearly independent and form a complete system in the W 1 2 (0, l). We will look for a solution to the problem (1) in the form of By substituting (10) in (11) we come to a system of ordinary differential equations: