Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems

Xiongxiong Bao; Wan-Tong Li

doi:10.3934/dcdsb.2020249

Article Contents

2021, Volume 26, Issue 7: 3621-3641. Doi: 10.3934/dcdsb.2020249

This issue Previous Article Finite-time cluster synchronization of coupled dynamical systems with impulsive effects Next Article Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory

Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems

Xiongxiong Bao^1, and
Wan-Tong Li^2, ,

1.
School of Science, Chang'an University, Xi'an, Shaanxi 710064, China
2.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

^* Corresponding author (wtli@lzu.edu.cn)
^* Corresponding author (wtli@lzu.edu.cn)

Received: December 31, 2019

Revised: June 30, 2020

Early access: August 2020

Published: July 2021

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Abstract

The current paper is devoted to the study of the existence and stability of generalized transition waves of the following time-dependent reaction-diffusion cooperative system

$ \begin{equation*} \frac{{\partial} \mathbf{u}}{{\partial} t}(t,x) = \Delta \mathbf{u}(t,x)+\mathbf{F}(t,\mathbf{u}(t,x)),\quad (x,t)\in {\mathbb{R}}^{N}\times {\mathbb{R}},\, \mathbf{u}\in \Bbb{R}^{K},\, K>1. \end{equation*} $

Here $ \mathbf{F}(t,\mathbf{u}(t,x)) $ depends on $ t\in\Bbb{R} $ in a general way. Recently, the spreading speeds and linear determinacy of the above time-dependent system have been studied by Bao et al. [J. Differential Equations 265 (2018) 3048-3091]. In this paper, using the principal Lyapunov exponent and principal Floquent bundle theory of linear cooperative systems, we prove the existence of generalized transition waves in any given direction with speed greater than the spreading speed by constructing appropriate subsolutions and supersolutions. When the initial value is uniformly bounded with respect to a weighted maximum norm, we further show that all solutions converge to the generalized transition wave solutions exponentially in time.

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Mathematics Subject Classification: 35B15, 35K55, 35K57, 92D25.

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