Extinction for fast diffusion equations with nonlinear sources

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 less than $m$ less than 1, in a bounded domain of $R^N$ with $N$ greater than 2. More precisely, we show that if p greater than m, the solution with small initial data vanishes in finite time, and if $p$ less than $m$, the maximal solution is positive for all $t$ greater than $0$. If $p=m$, then first eigenvalue of the Dirichlet problem plays a role.