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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