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Extinction for fast diffusion equations with nonlinear sources. (English) Zbl 1073.35134

Summary: We establish conditions for the extinction of solutions, in finite time, of the fast diffusion problem \(u_t=\Delta u^m+\lambda u^p\), \(0<m< 1\), in a bounded domain of \(\mathbb R^N\) with \(N> 2\). More precisely, we show that if \(p>m\), the solution with small initial data vanishes in finite time, and if \(p<m\), the maximal solution is positive for all \(t> 0\). If \(p=m\), then first eigenvalue of the Dirichlet problem plays a role.

MSC:

35K65 Degenerate parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
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