CRITICAL EXPONENT FOR A SYSTEM OF SLOW DIFFUSION EQUATIONS WITH BOTH REACTION AND ABSORPTION TERMS

Abstract

Let $\Omega$ is a bounded domain in $R^N$ with a smooth boundary $\partial\Omega$, $m,n\gt 1$ and $p,q,r,s,a,b$ are positive constants. For the initial and boundary value problem $$ \begin{array}{rl} u_t =\bigtriangleup u^m+v^p-au^r,& \quad x\in\Omega,\quad t\gt 0,\\ v_t =\bigtriangleup v^n+u^q-bv^s, & \quad x\in\Omega,\quad t\gt 0,\\ u=v=0, & \quad x\in\partial\Omega,\quad t\gt 0,\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x),& \quad x\in\Omega, \end{array} $$ we prove that all solutions are globally bounded if $pq \lt \max\{n, r\} \max\{n, s\}$; while there are finite time blowing up solutions if $pq \lt \max\{n, r\} \max\{n, s\}$ and the initial data are sufficiently large. For the critical case $pq=\max\{m,r\}$ $\max\{n,s\}$, the existence or nonexistence of global solutions depends on the relation between the exponents $m$, $n$, $r$, $s$, and also the range of the parameters $a$, $b$.