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$$L^ p$$-energy and blow-up for a semilinear heat equation. (English) Zbl 0631.35049
Nonlinear functional analysis and its applications, Proc. Summer Res. Inst., Berkeley/Calif. 1983, Proc. Symp. Pure Math. 45/2, 545-551 (1986).
The author considers the following initial boundary value problem: $u_ t(t,x)=\Delta u(t,x)+| u(t,x)|^{\gamma -1}u(t,x),\quad t>0,\quad x\in \Omega,$ $u(t,y)=0,\quad t>0,\quad y\in \partial \Omega \text{ and } u(0,x)=\phi(x),\quad x\in \Omega,$ where $$\Omega$$ is a bounded domain in $$R^ n$$ with smooth boundary and $$\gamma >1$$ is a fixed parameter.
A necessary condition on $$\phi$$ for the existence of a local solution is derived:
Theorem. Let $$\phi$$ be a positive Borel measure on $$\Omega$$ and u a nonnegative weak solution of the above problem on (0,T)$$\times \Omega$$. Then $(\gamma -1)t(e^{t\Delta}\phi)^{\gamma -1}\leq 1\text{ for all }t\in(0,T],\quad x\in \Omega.$ Furthermore it is proved that under certain circumstances the $$L^ p$$ solutions blow-up in $$L^ p(\Omega)$$ with $$p=n(\gamma -1)/2:$$
Theorem. Suppose $$p=n(\gamma -1)/2>1$$ and $$\phi \in L^ p(\Omega)$$. Let u: [0,T)$$\to L^ p(\Omega)$$ be the maximal continuous solution of the above problem and assume T finite. If $$p=2$$, or if $$2<p<\gamma +1$$ and u and $$u_ t$$ are both nonnegative throughout the trajectory, then $$\| u(t)\|_ p\to \infty$$ as $$t\to T$$.
Reviewer: S.Salsa

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35K15 Initial value problems for second-order parabolic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs