Semi-wavefront for a Belousov–Zhabotinskii reaction–diffusion system with spatio-temporal delay

Abstract

This paper considers a Belousov–Zhabotinskii reaction–diffusion system with spatio-temporal delay. The spatio-temporal delay is modeled as the convolution of $u(t,x)$ with a kernel function $g(t,x)\ge 0$, where ${\int }_R{\int }_0^{+\infty }g(s,y)dsdy=1$. By constructing an auxiliary system, applying Schauder’s fixed point theorem, and using a limiting argument, we demonstrate that the model admits non-negative traveling wave solutions connecting the equilibrium $(0,0)$ to some unknown positive steady states (also referred to as a semi-wavefront) when the wave speed satisfies $c\ge c^{\ast }=2\sqrt{1-r}$. Moreover, no such semi-wavefront exists if $c<c^{\ast }$. It is noteworthy that the kernel function is not required to have compact support in this analysis.