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.