Travelling wavefronts of Belousov–Zhabotinskii system with diffusion and delay

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China
Applied Mathematics Letters (Impact Factor: 1.34). 03/2009; 22(3):341-346. DOI: 10.1016/j.aml.2008.04.006
Source: DBLP


This paper is concerned with the existence, nonexistence and minimal wave speed of the travelling wavefronts of Belousov–Zhabotinskii system with diffusion and delay.

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    • "for the first condition and [26, Theorem 3.2] for the second one. The last conclusion is known from [30] (for h = 0) and [24] (for h ≥ 0). A very short elementary proof of it can be also found in Lemma 4.16. "
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    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 $h\geq 0$. 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 $h=0, r\not=1$, 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.
    Full-text · Article · Mar 2011 · Journal of Differential Equations
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    • "The traveling wave problems for reaction–diffusion systems without delay have been studied by many authors, see [6] [7]. Moreover, the research on traveling wave solutions to reaction–diffusion systems with delay have been widely studied in the literature [8] [9] [10] [11] [12] [13]. Schaaf [12] systematically studied two scalar reaction–diffusion equations with a single discrete delay by using the phase plane technique, the maximum principle for parabolic functional differential equations and the general theory of ordinary differential equations. "
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    ABSTRACT: This paper is concerned with a reaction–diffusion equation with time delay, which describes the dynamics of the blood cell production. The existence of the traveling wavefront is given by using the upper–lower solution technique and the monotone iteration.
    Preview · Article · Apr 2010 · Applied Mathematics Letters
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    ABSTRACT: Existence of travelling wave front solution is established for diffusive and competitive Lotka–Volterra system with delays. The approach used in this paper is the upper-lower solution technique and the monotone iteration. The same results are suitable to Belousov–Zhabotinskii model with delays and cooperative Lotka–Volterra system with delays.
    No preview · Article · Jun 2010 · Nonlinear Analysis Real World Applications
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