Research article

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

  • 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.

    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

    Related Papers:

  • 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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