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

Abstract

<p style='text-indent:20px;'>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 <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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&gt;1. \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>Here <inline-formula><tex-math id="M1">\begin{document}$ \mathbf{F}(t,\mathbf{u}(t,x)) $\end{document}</tex-math></inline-formula> depends on <inline-formula><tex-math id="M2">\begin{document}$ t\in\Bbb{R} $\end{document}</tex-math></inline-formula> in a general way. Recently, the spreading speeds and linear determinacy of the above time-dependent system have been studied by Bao et al. [<i>J. Differential Equations</i> 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.