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

Hui-Ling Niu; Hui-Ling Niu

doi:10.3934/math.2021020

AIMS Mathematics

2021, Volume 6, Issue 1: 314-332. doi: 10.3934/math.2021020

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Research article

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

Hui-Ling Niu ^,

School of Mathematics and Information Science, North Minzu University, Yinchuan, Ningxia 750021, People's Republic of China

Received: 16 August 2020 Accepted: 29 September 2020 Published: 12 October 2020
MSC : 35K57, 35C07, 35B35, 35B40

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.
- reaction-diffusion equation,
- nonlinear convection,
- V-shaped traveling front,
- multidimensional stability
Citation: Hui-Ling Niu. Multidimensional stability of V-shaped traveling fronts in bistable reaction-diffusion equations with nonlinear convection[J]. AIMS Mathematics, 2021, 6(1): 314-332. doi: 10.3934/math.2021020

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Abstract

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.

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