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Critical blow-up for quasilinear parabolic equations in exterior domains. (English) Zbl 0868.35064
The author studies critical blow-up for nonnegative solutions to the exterior Cauchy-Dirichlet problem for quasilinear parabolic equations. Precisely, let $$\Omega\subset\mathbb{R}^N$$, $$N\geq 2$$, be an exterior domain with a smooth boundary $$\partial\Omega$$, and consider the problem $\partial_tu=\Delta u^m+u^p\quad\text{in}\quad \Omega\times(0,T),\tag{$$*$$}$
$u(x,0)= u_0(x),\quad u(x,t)=0\quad\text{on}\quad \partial\Omega\times(0,T),$ with $$p>m\geq 1$$ and $$u_0(x)\geq 0$$. It is known that $$p_m= m+2/N$$ is a critical exponent for the problem $$(*)$$. That is, if $$m<p<p_m$$ then all nontrivial nonnegative weak solutions to $$(*)$$ blow-up in a finite time while, if $$p>p_m$$ then global solutions to $$(*)$$ exist for small enough initial data.
The paper under review deals with the critical case $$p=p_m$$. Namely, the author proves that all nonnegative nontrivial solutions to the problem $$(*)$$ blow-up in a finite time when $$N\geq 3$$ and $$p=p_m$$. The research is carried out by studying the asymptotic behaviour of solutions to the exterior Cauchy-Dirichlet problem for the equation $$\partial_tu=\Delta u^m$$.

##### MSC:
 35K65 Degenerate parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
##### Keywords:
exterior Cauchy-Dirichlet problem
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