*(English)*Zbl 0994.35027

The authors study blow-up and global existence for the system of porous medium equations

coupled through the nonlinear boundary conditions

The initial data $u(\xb7,0)$, $v(\xb7,0)$ are assumed to be continuous nonnegative and compactly supported, $m,n>1$, $p,q>0$. It is shown that all solutions of this problem exist globally if and only if $pq\le (m+1)(n+1)/4$. In the blow-up case, the authors find necessary and sufficient conditions in terms of $p,q,m,n$ for blow-up of all nontrivial solutions. They also establish the blow-up rate of blowing up solutions which are increasing in time, and they characterize the blow-up sets $B\left(u\right)$, $B\left(v\right)$ of solutions satisfying the blow-up rate mentioned above. Each of $B\left(u\right)$, $B\left(v\right)$ is either $\left\{0\right\}$, a bounded interval containing 0 or the interval $[0,\infty )$ and any combination of these alternatives is possible.