Adapting the time-step to recover the asymptotic behavior in a blow-up problem

The equation $u_t=\mathrm{\Delta }u+u^p$ with homegeneous Dirichlet boundary conditions has solutions with blow-up if $p>1$. An adaptive time-step procedure is given to reproduce the asymptotic behvior of the solutions in the numerical approximations. We prove that the numerical method reproduces the blow-up cases, the blow-up rate and the blow-up time. We also localize the numerical blow-up set.