Semidiscretization in time of nonlinear parabolic equations with blowup of the solution

In this paper the author considers the first boundary value problem for the nonlinear equation $u_t - \Delta u^m = \alpha u^m $ in $\Omega $, a smooth bounded domain in $\mathbb{R}^n $ with the zero lateral boundary condition and with a positive initial condition; m is supposed to be larger than one and $\alpha $ positive. A scheme for the discretization in time of that problem is proposed. It is proven that if the exact solution blows up in a finite time, it is the same for the numerical solution. Estimates of the blow-up time are obtained. The stability of the method and the convergence for a class of initial conditions is proved.