Long-time behavior of a cooperative periodic-parabolic system in a spatiotemporally degenerate environment

This paper is the continuation of our work [1] and [41] on the study of the cooperative periodic-parabolic system:{∂tu−Δu=μu+α(x,t)v−a(x,t)up∂tv−Δv=μv+β(x,t)u−b(x,t)vqinΩ×(0,∞),(∂νu,∂νv)=(0,0)on∂Ω×(0,∞),(u(x,0),v(x,0))=(u0(x),v0(x))>(0,0)inΩ, where p,q>1p,q>1, Ω⊂RNΩ⊂RN (N≥2N≥2) is a bounded smooth domain, α,β>0α,β>0 and a,b≥0a,b≥0 are smooth functions that are T-periodic in t, and μ   is a varying parameter. The unknown functions u(x,t)u(x,t) and v(x,t)v(x,t) stand for the densities of two cooperative species at location x and time t  . The aim of our work is to establish the long-time behavior of (u,v)(u,v) when the species are exposed to a spatiotemporally degenerate environment. In [1] and [41], we dealt with the three cases that a and b have simultaneous spatiotemporal degeneracy, simultaneous spatial degeneracy and simultaneous temporal degeneracy. In this paper we consider some other natural situations of degeneracies. Our results reveal further interesting effects of spatial and temporal degeneracies on the dynamics of such a cooperative system. In addition, we provide a sharp improvement of the results in [41] when simultaneous temporal degeneracy occurs.