Multizonal Internal Layers in a Stationary Piecewise–Smooth Reaction-Diffusion Equation in the Case of the Difference of Multiplicity for the Roots of the Degenerate Solution

Pub Date : 2024-06-13 DOI:10.1134/s0965542524700179
Qian Yang, Mingkang Ni
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Abstract

A singularly perturbed stationary problem for a one-dimensional reaction-diffusion equation in the case when the degenerate equation has multiple roots is studied. This is a new class of problems with discontinuous reactive terms along some curve that is independent of the small parameter. The existence of a smooth solution with the transition from the triple root of one degenerate equation to the double root of the other degenerate equation in the neighborhood of some point on the discontinuous curve is studied. Based on the existence theorem of classical boundary value problems and the technique of matching asymptotic expansion, the existence of a smooth solution is proved. And the point itself and the asymptotic representation of this solution are constructed by the matching technique and modified boundary layer function method.

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稳态片状光滑反应-扩散方程中的多层内层与退化解根的多重性差异
摘要 研究了一维反应-扩散方程在退化方程有多个根的情况下的奇异扰动静止问题。这是一类新问题,其反应项沿着与小参数无关的曲线不连续。研究了在不连续曲线上的某一点附近,是否存在从一个退化方程的三重根过渡到另一个退化方程的双重根的平稳解。基于经典边界值问题的存在定理和匹配渐近展开技术,证明了平稳解的存在性。并通过匹配技术和修正边界层函数法构建了点本身和该解的渐近表示。
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