A priori bounds and complete blow up of positive solutions of indefinite superlinear parabolic problems. (English) Zbl 1071.35026

Summary: We study a priori estimates of positive solutions of the equation \[ \partial_t u-\Delta u=\lambda u+ a(x)u^p,\quad x\in\Omega,\quad t> 0, \] satisfying the homogeneous Dirichlet boundary conditions. Here \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), \(\lambda\in \mathbb{R}\), \(p> 1\) is subcritical, \(a\in C(\overline\Omega)\) changes sign and \(a\), \(p\) satisfy some additional technical hypotheses. Assume that the solution \(u\) blows up in a finite time \(T\) and the set \(\Omega^+ := \{x\in\Omega: a(x)> 0\}\) is connected. Using our a priori bounds, we show that u blows up completely in \(\Omega^+\) at \(t= T\) and the blow up time \(T\) depends continuously on the initial data.


35B45 A priori estimates in context of PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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