Solvability of a Pseudohyperbolic Equation with a Nonlinear Boundary Condition

This paper is devoted to the fundamental problem of investigating the solvability of an initial-boundary value problem for a quasi-linear pseudo-hyperbolic equation (also called Sobolev type equations) with a sufficiently smooth boundary. In this work, we study an initial-boundary value problem for a quasi-linear pseudo-hyperbolic equation with a nonlinear Neumann-Dirichlet boundary condition. The paper uses the Galerkin method to prove the existence of a weak solution of a quasi-linear pseudo-hyperbolic equation in a bounded domain. Using Sobolev embedding theorems, priori estimates of the solution are obtained. The use of Galerkin approximations allows us to obtain an overtime estimate of the solution's existence. A local theorem on the existence of a weak generalized solution is proved. A priori estimates and the Rellich-Kondrashov theorem are used to prove the existence of the desired solution to the boundary value problem under consideration. The uniqueness of a weak generalized solution to the initial boundary value problem of a quasi-linear pseudo-hyperbolic equation is proved on the basis of the obtained a priori estimates and the application of the Gronwall-Bellman Lemma. The need to consider and study such initial-boundary value problems for a quasi-linear pseudo-hyperbolic equation follows from practical needs. For example, when solving differential equations that model physical processes, it is important that there is a good match between the selected model and the real object.