Extinction for fast diffusion equations with nonlinear sources

Abstract

We establish conditions for the extinction of solutions, in finite time, of the fast diffusion problem ut = ∆um + λup, 0 < m < 1, in a bounded domain of RN 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.