  The equation $u_t = \Delta u + u^p$ 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.
