Traveling waves for a model of the Belousov-Zhabotinsky reaction

Abstract

Following J.D. Murray, we consider a system of two differential equations that models traveling fronts in the Noyes-Field theory of the Belousov-Zhabotinsky (BZ) chemical reaction. We are also interested in the situation when the system incorporates a delay $h\ge 0$. As we show, the BZ system has a dual character: it is monostable when its key parameter $r\in (0,1]$ and it is bistable when $r>1$. For $h=0,r\ne 1$, and for each admissible wave speed, we prove the uniqueness of monotone wavefronts. Next, a concept of regular super-solutions is introduced as a main tool for generating new comparison solutions for the BZ system. This allows to improve all previously known upper estimations for the minimal speed of propagation in the BZ system, independently whether it is monostable, bistable, delayed or not. Special attention is given to the critical case $r=1$ which to some extent resembles to the Zeldovich equation.

23 pages, to appear in the Journal of Differential Equations

References in corpus (3)

• On uniqueness of semi-wavefronts (Diekmann-Kaper theory of a nonlinear convolution equation re-visited)
• Monotone traveling wavefronts of the KPP-Fisher delayed equation
• Bistable Traveling Waves for Monotone Semiflows with Applications

Cited by in corpus (4)

• Travelling waves for a bistable reaction-diffusion equation with delay
• Traveling waves in the nonlocal KPP-Fisher equation: different roles of the right and the left interactions
• Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction
• Two reasons for the appearance of pushed wavefronts in the Belousov-Zhabotinsky system with spatiotemporal interaction