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Asymptotic behavior of solutions for systems of periodic reaction-diffusion equations in unbounded domains. (English) Zbl 1122.35059

Summary: The purpose of this paper is to study the asymptotic behavior of the time-dependent solutions to a coupled system of reaction-diffusion equations with periodic coefficients in unbounded domains, including the whole space \(\mathbb R^n\), the exterior \(\Omega_e\) of a bounded domain and the half-space \(\mathbb R^n_+=\{x=(x_1,\dots,x_n)\in \mathbb R^n\); \(x_n>0\}\), under Dirichlet boundary conditions. It is shown that if the reaction function in the system possesses a mixed quasimonotone property and the corresponding periodic system has a pair of coupled upper and lower solutions, then the periodic system has a pair of quasi-solutions and the sector between the quasi-solutions is an attractor of the system of periodic reaction-diffusion equations. Under some additional conditions this attractor leads to the asymptotic stability of a periodic solution for the periodic system. The approach to the problem is based on the method of upper and lower solutions, a ladder argument and a bootstrap technique. Finally, a competitor-competitor-mutualist model is given to illustrate the obtained results.

MSC:

35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35B10 Periodic solutions to PDEs
92D25 Population dynamics (general)
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
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