Abstract

The existence of time-periodic solutions of the boundary-layer type to a two-dimensional reaction–diffusion problem with a small-parameter coefficient of a parabolic operator is proved in the case of singularly perturbed boundary conditions of the second kind. An asymptotic approximation with respect to the small parameter is constructed for these solutions. The set of boundary conditions for which these solutions exist is studied and the local uniqueness and asymptotic Lyapunov stability are established for them. It is shown that, unlike the analogous Dirichlet problem, for which such a solution is unique, there can be several solutions of this kind for the problem under consideration, each of which has its domains of stability and local uniqueness. To prove these facts, results based on the asymptotic principle of differential inequalities are used.Keywords: , , , , , , .

中文翻译：

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第二类边界条件奇摄动的二维反应扩散问题边界层周期解的存在性和稳定性

摘要

在第二类奇异摄动边界条件下，证明了具有小参数系数的抛物线算子对二维反应扩散问题的边界层类型的时间周期解的存在。对于这些解决方案，构造了一个关于小参数的渐近近似。研究了存在这些解的边界条件集，并为其建立了局部唯一性和渐近Lyapunov稳定性。结果表明，与类似的狄利克雷问题不同，对于此类问题，这种解决方案是唯一的，对于正在考虑的问题，可以有几种此类解决方案，每种解决方案都具有其稳定性和局部唯一性。为了证明这些事实，关键字：，，，，，。