Multidimensional stability of V-shaped traveling fronts in bistable reaction-diffusion equations with nonlinear convection

• Published in: AIMS mathematics Vol. 6; no. 1; pp. 314 - 332
• Main Author: Niu, Hui-Ling
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2021
• Subjects: multidimensional stability; nonlinear convection; reaction-diffusion equation; v-shaped traveling front
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2021020

This paper is concerned with the multidimensional stability of V-shaped traveling fronts for a reaction-diffusion equation with nonlinear convection term in $\mathbb{R}^n$ ($n\geq3$). We consider two cases for initial perturbations: one is that the initial perturbations decay at space infinity and another one is that the initial perturbations do not necessarily decay at space infinity. In the first case, we show that the V-shaped traveling fronts are asymptotically stable. In the second case, we first show that the V-shaped traveling fronts are also asymptotically stable under some further assumptions. At the same time, we also show that there exists a solution that oscillates permanently between two V-shaped traveling fronts, which means that the traveling fronts are not asymptotically stable under general bounded perturbations.