Abstract

We prove the existence and study the stability of time-periodic boundary layer solutions for a two-dimensional reaction–diffusion problem with a small parameter multiplying the parabolic operator for the case of singularly perturbed boundary conditions of the third kind. An asymptotic approximation to such solutions with respect to the small parameter is constructed. Conditions under which these solutions are Lyapunov asymptotically stable, as well as conditions under which such solutions are unstable, are obtained. For the proof, we used results based on applying the asymptotic method of differential inequalities and the Krein–Rutman theorem.

中文翻译：

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具有第三类奇异摄动边界条件的反应扩散问题的周期边界层解

摘要

我们证明了存在，并研究了在第三类奇异摄动边界条件下，带有小参数的二维反应扩散问题乘以抛物线算子的时间周期边界层解的稳定性。构造了关于小参数的此类解的渐近近似。获得了这些解为Lyapunov渐近稳定的条件，以及这些解为不稳定的条件。为了证明这一点，我们使用了基于微分不等式渐近方法和Krein-Rutman定理的结果。