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Blowup and life span of solutions for a semilinear parabolic equation. (English) Zbl 0909.35056
The authors consider the following Cauchy problem $$u_{t}=\Delta u +| u| ^{p-1}u\quad \text{in } {\bbfR}^{N}\times (0,\infty),\quad u(x,0)=u_{0}(x)\quad \text{in } {\bbfR}^{N},$$ with $p>1$. Set $\Omega= \{(r,\omega)\in {\bbfR}^{+}\times S^{N-1}: r>R$, $d(\omega,\omega_{0})< cr^{-\mu}\}$ for some $R>0$, $c>0$, $\omega_{0}\in S^{N-1}$ and $0\leq \mu<1$ with $d(\cdot,\cdot)$ being the standard distance on $S^{N-1}$. The authors show that if $u_{0}$ decays like $| x| ^{-\alpha}$ as $| x| \to\infty$ in $\Omega$ with $0<\alpha<2(1-\mu)/(p-1)$, then the solution of the above problem blows up in a finite time regardless of the behaviour of $u_{0}$ outside $\Omega$. Later on, the life span of such a solution with $u_{0}=\lambda\varphi$ is estimated from above for small values of $\lambda>0$ in terms of $p$, $\alpha$ and $\mu$.

35K15Second order parabolic equations, initial value problems
35B40Asymptotic behavior of solutions of PDE
35K57Reaction-diffusion equations
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