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

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)].

MSC:

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

Citations:

Zbl 0819.35076
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