Traveling wave solution for a nonlocal Belousov-Zhabotinskii reaction-diffusion system

Abstract

We consider a Belousov-Zhabotinskii reaction-diffusion system with nonlocal effects and study the existence of traveling wave solutions. By constructing appropriate super- and sub-solutions and using Schauder’s fixed point theorem, we show that there is a critical speed c * > 0 such that when the wave speed c > c *, there exists a traveling wave solution connecting (0, 0) to a positive steady-state, while there is no traveling wave solution when c < c *. Moreover, we also examine a special case where ϕ 1(x) is the Dirac function, and demonstrate the existence of the traveling wave solution connecting the equilibria (0, 0) and (1, 1) for c > c *, whereas the traveling wave solution does not exist when c < c *. Finally, the long-time behavior of the solution is investigated through numerical simulation and theoretical analysis, and it is found that the choice of kernel functions and the setting of initial value conditions play a crucial role.