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 - Δu m = αu p 1 in Ω, a smooth bounded domain in R d (d > 1) with the zero lateral boundary condition and with a positive initial condition, m E ]0,1[ (fast diffusion problem), α > 0 and p1 > 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.