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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
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