Solvability of a Pseudohyperbolic Equation with a Nonlinear Boundary Condition

  • S. E. Aitzhanov Al-Farabi Kazakh National University, Almaty, Kazakhstan; Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan http://orcid.org/0000-0001-5877-7195
  • К. S. Bekenaeva Abai Kazakh National Pedagogical University, Almaty, Kazakhstan
  • G. О. Zhumagul Abai Kazakh National Pedagogical University, Almaty, Kazakhstan

Abstract

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.

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How to Cite
AITZHANOV, S. E.; BEKENAEVA, К. S.; ZHUMAGUL, G. О.. Solvability of a Pseudohyperbolic Equation with a Nonlinear Boundary Condition. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 108, n. 4, p. 26-37, dec. 2020. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/817>. Date accessed: 20 jan. 2021. doi: https://doi.org/10.26577/JMMCS.2020.v108.i4.03.
Keywords pseudohyperbolic equations, nonlinear boundary conditions, Galerkin method, existence of a solution, uniqueness of a solution