Periodic Contrast Structures in the Reaction–Diffusion–Advection Equation with a KPZ Nonlinearity

E. I. Nikulin, A. O. Orlov

Lomonosov Moscow State University

Abstract: A singularly perturbed time-periodic boundary-value problem for a parabolic reaction–advection–diffusion equation with a nonlinearity containing the squared gradient of the unknown function (KPZ nonlinearity) is studied. A periodic solution with an internal transition layer is considered in the noncritical and critical cases. An asymptotic approximation of the solution is constructed, and the asymptotic behavior of a point of the transition layer is determined. Existence theorems and asymptotic stability are proved by the method of differential inequalities.

Keywords: reaction–advection–diffusion equation, KPZ nonlinearity, method of differential inequalities, internal transition layer, small parameter, periodic problem.

UDC: 517.9

MSC: 35K61

Received: 09.05.2024
Revised: 06.08.2024

DOI: 10.4213/mzm14361

 English version: Mathematical Notes, 2025, 118:2, 321–337

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