Abstract
We study the asymptotic behavior of the nonnegative solutions of a periodic reaction diffusion system. By obtaining a priori upper bound of the nonnegative periodic solutions of the corresponding periodic diffusion system, we establish the existence of the maximum periodic solution and the asymptotic boundedness of the nonnegative solutions of the initial boundary value problem.
Highlights
We study the asymptotic behavior of the nonnegative solutions of a periodic reaction diffusion system
By obtaining a priori upper bound of the nonnegative periodic solutions of the corresponding periodic diffusion system, we establish the existence of the maximum periodic solution and the asymptotic boundedness of the nonnegative solutions of the initial boundary value problem
B2uα[2] vβ2, x, t ∈ Ω × R, 1.2 with initial boundary conditions u x, t v x, t 0, x, t ∈ ∂Ω × R, 1.3 u x, 0 u0 x, v x, 0 v0 x, x ∈ Ω, 1.4 where m1, m2 > 1, α1, α2, β1, β2 ≥ 1, Ω ⊂ Rn is a bounded domain with a smooth boundary ∂Ω, b1 b1 x, t and b2 b2 x, t are nonnegative continuous functions and of T -periodic T > 0 with respect to t, and u0 and v0 are nonnegative bounded smooth functions
Summary
By obtaining a priori upper bound of the nonnegative periodic solutions of the corresponding periodic diffusion system, we establish the existence of the maximum periodic solution and the asymptotic boundedness of the nonnegative solutions of the initial boundary value problem
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