Abstract

The work is devoted to the fundamental problem of studying the solvability of the initial-boundary value problem for a pseudo-parabolic equation (also called Sobolev type equations) with a fairly smooth boundary. In this paper, the initial-boundary value problem for a quasilinear equation of a pseudoparabolic type with a nonlinear Neumann–Dirichlet boundary condition is studied. From a physical point of view, the initial-boundary-value problem we are considering is a mathematical model of quasi-stationary processes in semiconductors and magnetics, taking into account the most diverse physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions of tasks boundary conditions of which are linear with respect to the function and its derivatives. Among these methods, Galerkin’s method leads to the simplest calculations. In the paper, by means of the Galerkin method the existence of a weak solution of a pseudoparabolic equation in a bounded domain is proved. The use of the Galerkin approximations allows us to get an estimate above the time of the solution existence. Using Sobolev ’s attachment theorem, a priori solution estimates are obtained. The local theorem of the existence of the solution has been proved. The uniqueness of the weak generalized solution of the initial-boundary value problem of quasi-linear equations of pseudoparabolic type is proved on the basis of a priori estimates.A special place in the theory of nonlinear equations is taken by the range of studies of unlimited solutions, or, as they are otherwise called, modes with exacerbation. Nonlinear evolutionary problems that allow unlimited solutions are globally intractable: solutions increase indefinitely over a finite period of time. Sufficient conditions have been obtained for the destruction of its solution over finite time in a limited area with a nonlinear Neumann–Dirichle boundary condition.

中文翻译：

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具有非线性边界条件的伪抛物方程的可解性

摘要

这项工作致力于研究具有相当光滑边界的拟抛物线方程（也称为Sobolev型方程）的初边值问题的可解性的基本问题。在本文中，研究了具有非线性Neumann-Dirichlet边界条件的拟抛物型拟线性方程的初边值问题。从物理角度来看，考虑到最多种物理因素，我们正在考虑的初边值问题是半导体和磁性材料中准平稳过程的数学模型。许多近似方法适用于查找任务边界条件的特征值和特征函数，这些任务的边界条件相对于该函数及其导数是线性的。在这些方法中，Galerkin的方法导致最简单的计算。在本文中，通过Galerkin方法，证明了在有界域中伪抛物方程的弱解的存在。Galerkin近似的使用使我们能够获得高于解存在时间的估计。使用Sobolev的附着定理，可以获得先验解估计。证明了解存在的局部定理。在先验估计的基础上，证明了伪抛物型拟线性方程组初边值问题的弱广义解的唯一性。在非线性方程组理论中，无穷范围的研究占有特殊地位。解决方案，或称为加重模式。允许无穷解的非线性演化问题在全球都是棘手的：解在有限的时间内无限增加。在非线性诺伊曼-狄里克尔边界条件下，已经获得了在有限时间内在有限时间内破坏其解的充分条件。