Totally discrete explicit and semi-implicit Euler methods for a blow-up problem in several space dimensions

Abstract

The equation u t =Δu+u p with homogeneous Dirichlet boundary conditions has solutions with blow-up if p>1. An adaptive time-step procedure is given to reproduce the asymptotic behavior of the solutions in the numerical approximations. We prove that the numerical methods reproduce the blow-up cases, the blow-up rate and the blow-up time. We also localize the numerical blow-up set.