Semi-discretization in time of a fast diffusion equation

Abstract

In this paper, we consider the first boundary value problem for the nonlinear concentration dependent diffusion equation: u′(t) − Δu^m(t) = 0 in Ω, a smooth bounded domain in $\text{R}$ⁿ, with the zero lateral boundary condition and with a positive initial condition; m is supposed to be between 0 and 1. Then the solution decays to zero in some finite time $\text{T}{}^{\mathrm{\ast }}$ depending upon the initial data. Here we propose a scheme for the discretization in time of that problem; we prove that the numerical solution is null after a finite number of time steps and we obtain the convergence of the method and estimates of the numerical extinction time.