Qualitative Analysis of a Cooperative Reaction-Diffusion System in a Spatiotemporally Degenerate Environment

Abstract

In this paper, we are concerned with the cooperative system in which ∂tu − Δu = μu + α(x, t)v − a(x, t)up and ∂tv − Δv = μv + β(x, t)u − b(x, t)vq in Ω × (0,∞); (∂ν u, ∂νv) = (0, 0) on ∂Ω×(0,∞); and (u(x, 0), v(x, 0)) = (u0(x), v0(x)) > (0, 0) in Ω, where p, q > 1, Ω ⊂ RN (N ≥ 2) is a bounded smooth domain, α, β > 0 and a, b ≥ 0 are smooth functions that are T-periodic in t, and μ is a varying parameter. The unknown functions u(x, t) and v(x, t) represent the densities of two cooperative species. We study the long-time behavior of (u, v) in the case that a and b vanish on some subdomains of Ω × [0, T]. Our results show that, compared to the nondegenerate case where a, b > 0 on Ω × [0, T], such a spatiotemporal degeneracy can induce a fundamental change to the dynamics of the cooperative system.