Algebra, analysis and probability for a coupled system of reaction- diffusion equations. (English) Zbl 0824.60070

Champneys, Alan; Harris, Simon; Toland, John; Warren, Jonathan; Williams, David

Philos. Trans. R. Soc. Lond., Ser. A 350, No. 1692, 69-112 (1995).

The authors concentrate on the problems of a coupled system of reaction- diffusion equations \[ \partial u_ 1/ \partial t ={1\over 2}a_ 1 \partial u^ 2_ 1/ \partial x^ 2 + r_ 1 (u^ 2_ 1 - u_ 1)- \theta q_ 1 u_ 1 + \theta q_ 1 u_ 2, \] and \[ \partial u_ 2/ \partial t ={1\over 2}a_ 2 \partial^ 2 u_ 2/ \partial x^ 2 + r_ 2 (u^ 2_ 2 - u_ 2) + \theta q_ 2 u_ 1 - \theta q_ 2 u_ 2, \] where \(a_ i\), \(r_ i\), \(q_ i\) are fixed positive constants, \(\theta\) is a positive parameter, and \(u_ i = u_ i (t,x)\) \((i = 1,2)\). The authors provide a complete treatment of the existence, uniqueness, and asymptotic behavior of the monotone traveling-wave solutions (i.e., \(u_ i = w_ i (x - ct)\), where \(w_ i\) is a monotone function on \(R\) with \(w_ i (- \infty) = 0\) and \(w_ i (+ \infty) = 1)\) to various equilibria, both by differential equation theory using the maximum principle and by probability theory using martingale technique. The proofs of the existence but not uniqueness were also developed by others using other methods [e.g., cf. A. I. Volpert and V. A. Volpert, Commun. Partial Differ. Equations 18, No. 12, 2051-2069 (1993; Zbl 0819.35076)].

Reviewer: Wu Chengxun (Shanghai)

MSC:

60H30	Applications of stochastic analysis (to PDEs, etc.)
35K57	Reaction-diffusion equations
60G46	Martingales and classical analysis
60G44	Martingales with continuous parameter

Keywords:

reaction-diffusion equations; existence; uniqueness; asymptotic behavior; maximum principle; martingale technique

Citations:

Zbl 0819.35076