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)≥0, where ∫R∫0+∞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≥c∗=21−r. Moreover, no such semi-wavefront exists if c<c∗. It is noteworthy that the kernel function is not required to have compact support in this analysis.