An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

Antontsev S.N. Aitzhanov S.E. Zhanuzakova D.T.
15 January 2023 John Wiley and Sons Ltd

Mathematical Methods in the Applied Sciences
2023 #46 Issue 1 1111 - 1136 pp.

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 quasistationary 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. On the basis of a priori estimates, we prove a local existence theorem and 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.

asymptotic behavior of the solution , blow-up of the solution , Galerkin method , nonlinear boundary conditions , pseudoparabolic equations , the existence of a solution , uniqueness of the solution

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Lavrentyev Institute of Hydrodynamics of SB RAS, Novosibirsk, Russian Federation
CMAF-CIO, University of Lisbon, Lisbon, Portugal
Al-Farabi Kazakh National University, Almaty, Kazakhstan

Lavrentyev Institute of Hydrodynamics of SB RAS
CMAF-CIO
Al-Farabi Kazakh National University

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