## 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.

### MSC:

 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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