Lower bounds for blow-up time in parabolic problems under Dirichlet conditions (Q864635)

This paper is devoted to the study of the semilinear heat equation \(u_t=\Delta u+f(u)\) under homogeneous Dirichlet boundary conditions and appropriate constraints on the nonlinear term \(f(u)\). The authors impose conditions which insure that a solution exists locally, but the solution may possibly blow up at some finite time \(t^*\). By means of a first order differential inequality, the authors determine a lower bound for the blow-up time \(t^*\). An alternative method for this approach, based essentially on a comparison principle, is also proposed in the present paper, under slightly altered conditions on \(f\).