Abstract

This paper considers an initial-boundary value problem for a reaction–diffusion equation with a singularly perturbed Neumann boundary condition in a closed, simply connected two-dimensional domain. From a physical point of view, the problem describes processes with an intensive flow through the boundary of a given area. The existence of a stationary solution is proved, its asymptotic is constructed, and the Lyapunov stability conditions for it are established. The asymptotics of the solution are constructed by the classical Vasilieva algorithm using the Lusternik–Vishik method. The existence and stability of the solution are proved using the asymptotic method of differential inequalities.

中文翻译：

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具有奇摄动Neumann边界条件的反应扩散方程边界层平稳解的存在性

摘要

本文考虑了在封闭，简单连接的二维域中具有奇摄动Neumann边界条件的反应扩散方程的初边值问题。从物理角度看，问题描述的是密集流过给定区域边界的过程。证明了平稳解的存在性，构造了它的渐近性，并建立了其的Lyapunov稳定性条件。解决方案的渐近性是通过经典的Vasilieva算法使用Lusternik-Vishik方法构造的。用微分不等式的渐近方法证明了解的存在性和稳定性。