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Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions. (English) Zbl 0994.35027

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

${u}_{t}={\left({u}^{m}\right)}_{xx},\phantom{\rule{1.em}{0ex}}{v}_{t}={\left({v}^{n}\right)}_{xx},\phantom{\rule{2.em}{0ex}}x>0,\phantom{\rule{4pt}{0ex}}t>0,$

coupled through the nonlinear boundary conditions

$-{\left({u}^{m}\right)}_{x}\left(0,t\right)={v}^{p}\left(0,t\right),\phantom{\rule{1.em}{0ex}}-{\left({v}^{n}\right)}_{x}\left(0,t\right)={u}^{q}\left(0,t\right),\phantom{\rule{2.em}{0ex}}t>0·$

The initial data $u\left(·,0\right)$, $v\left(·,0\right)$ 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 $pq\le \left(m+1\right)\left(n+1\right)/4$. 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 $\left[0,\infty \right)$ and any combination of these alternatives is possible.

##### MSC:
 35B33 Critical exponents (PDE) 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35B35 Stability of solutions of PDE 35B40 Asymptotic behavior of solutions of PDE 35K60 Nonlinear initial value problems for linear parabolic equations
##### Keywords:
porous medium equation; blow-up rate; blow-up sets