Abstract

We study the problem of the existence and asymptotic stability of a stationary solution of an initial boundary value problem for the reaction–diffusion–advection equation assuming that the reaction and advection terms are comparable in size and have a jump along a smooth curve located inside the studied domain. The problem solution has a large gradient in a neighborhood of this curve. We prove theorems on the existence, asymptotic uniqueness, and Lyapunov asymptotic stability for such solutions using the method of upper and lower solutions. To obtain the upper and lower solutions, we use the asymptotic method of differential inequalities that consists in constructing them as modified asymptotic approximations in a small parameter of solutions of these problems. We construct the asymptotic approximation of a solution using a modified Vasil’eva method.

中文翻译：

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具有不连续反应和对流项的二维边界值反应-扩散-平流问题的内部过渡层解

摘要

我们研究了反应-扩散-平流方程初始边值问题的平稳解的存在性和渐近稳定性问题，假设反应项和对流项的大小相当，并且沿着位于方程内部的平滑曲线跳跃。研究领域。问题解在这条曲线的邻域内有一个很大的梯度。我们使用上下解的方法证明了此类解的存在性、渐近唯一性和Lyapunov渐近稳定性定理。为了获得上下解，我们使用微分不等式的渐近方法，该方法包括将它们构造为这些问题的解决方案的小参数中的修正渐近近似。