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An Initial Boundary Value Problem for a Pseudoparabolic Equation with a Nonlinear Boundary Condition
  • Serik Aitzhanov,
  • Stanilslav Antontsev,
  • Dinara Zhanuzakova
Serik Aitzhanov
Al-Farabi Kazakh National University

Corresponding Author:[email protected]

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Stanilslav Antontsev
FSBIS Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences
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Dinara Zhanuzakova
Al-Farabi Kazakh National University
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An initial-boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann-Dirichlet type is investigated in this work. From a physical point of view, the initial-boundary value problem considered here is a mathematical model of quasi-stationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. In this article, by the Galerkin method to prove the existence of a weak solution to the initial-boundary value problem for a pseudoparabolic equation in a bounded domain. On the basis of a priori estimates, we prove a local existence theoremand uniqueness for a weak generalized solution of the initial-boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann-Dirichlet boundary condition are obtained.
15 Mar 2022Submitted to Mathematical Methods in the Applied Sciences
16 Mar 2022Submission Checks Completed
16 Mar 2022Assigned to Editor
05 Apr 2022Reviewer(s) Assigned
17 Jun 2022Review(s) Completed, Editorial Evaluation Pending
01 Jul 2022Editorial Decision: Accept