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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 =Δu m +λu p , 0<m<1, in a bounded domain of 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:
35K65Parabolic equations of degenerate type
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35K20Second order parabolic equations, initial boundary value problems
35K55Nonlinear parabolic equations