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 $\partial_tu-\Delta u=\mu u+\alpha(x,t)v-a(x,t)u^p$ and $\partial_tv-\Delta v=\mu v+\beta(x,t)u-b(x,t) v^q$ in $\Omega\times (0,\infty)$; $(\partial_\nu u,\partial_\nu v)=(0,0)$ on $\partial\Omega\times(0,\infty)$; and $(u(x,0),v(x,0))=(u_0(x),v_0(x))> (0,0)$ in $\Omega$, where $p,\,q>1$, $\Omega\subset\mathbb{R}^N\; (N\geq 2)$ is a bounded smooth domain, $\alpha,\,\beta>0$ and $a,\, b\geq 0$ are smooth functions that are $T$-periodic in $t$, and $\mu$ 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 $\Omega\times[0,T]$. Our results show that, compared to the nondegenerate case where $a,\,b>0$ on $\overline\Omega\times[0,T]$, such a spatiotemporal degeneracy can induce a fundamental change to the dynamics of the cooperative system.

Locations

• e-Archivo (Carlos III University of Madrid) - View - PDF
• SIAM Journal on Mathematical Analysis - View