Numerical solution of a fast diffusion equation

In this paper, the authors consider the first boundary value problem for the nonlinear reaction diffusion equation: $u_{t}-\Delta u^{m}=\alpha u^{p_{1}}$ in $\Omega$, a smooth bounded domain in $\mathbb{R}^{d} (d\geq 1)$ with the zero lateral boundary condition and with a positive initial condition, $m\in ]0,1[$ (fast diffusion problem), $\alpha \geq 0$ and $p_{1}\geq m$. Sufficient conditions on the initial data are obtained for the solution to vanish or become infinite in a finite time. A scheme for the discretization in time of this problem is proposed. The numerical scheme preserves the essential properties of the initial problem; namely existence of an extinction or a blow-up time, for which estimates have been obtained. The convergence of the method is also proved.