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Existence of travelling wave solutions for reaction-diffusion-convection systems via the Conley index theory. (English) Zbl 1109.35364

The existence of monotone traveling waves for a rather general class of reaction-diffusion-convection systems is proved via Conley index theory. The assumptions on the coefficients include a local monotonicity condition of the source term. The article extends previous results by A. I. Volpert, Vit. A. Volpert and Vl. A. Volpert [Traveling wave solutions of parabolic systems. Translations of Mathematical Monographs. 140. Providence, RI: American Mathematical Society (1994; Zbl 1001.35060)] and by E. C. M. Crooks and J. F. Toland [Topol. Methods Nonlinear Anal. 11, No. 1, 19–43 (1998; Zbl 0920.35075)]. The proof of the main result, which comprises almost the whole paper, is broken down in a series of lemmas. The results are specified for ecological systems of mutualist type. As an application, conditions for the existence of ionization waves in a multicomponent plasma are given.

MSC:

35K57 Reaction-diffusion equations
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
37B30 Index theory for dynamical systems, Morse-Conley indices
37C29 Homoclinic and heteroclinic orbits for dynamical systems
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