Asymptotic Behavior of Solution for Coupled Reaction Diffusion System by Order m

Abstract

The aim of this paper is to prove that asymptotic behavior in the time of solutions for the weakly coupled reaction diffusion system:∂ui/∂t − di∆ui = fi (u1, u2, …, um) in Ω×R+,∂ui/∂η = 0 in ∂Ω×R+, (0.1)ui(., 0) = ui0(.) in Ω,where Ω is an open bounded domain of class C1 in Rn, ui(t, x), i=1, m, t≥0, x∈Ω are real valued functions. We treat the system (0.1) as a dynamical system in C(Ω) × C(Ω) × ... × C(Ω) and apply Lyapunov type stability techniques. A key ingredient in this analysis is a result which establishes that the orbits of the dynamical system are precompact in C(Ω) × C(Ω) × ... × C(Ω). As a consequence of Arzela-Ascoli theorem, this will be satisfied if the orbits are, for example, uniformly bounded in C1(Ω) × C1(Ω) × ... × C1(Ω) for t>0.