Lower bounds for blow-up time in a nonlinear parabolic problem

For a parabolic problem with a gradient nonlinearity which was introduced by Chipot and Weissler [M. Chipot, F.B. Weissler, Some blow up results for a nonlinear parabolic problem with a gradient term, SIAM J. Math. Anal. 20 (1989) 886–907] (see also [B. Kawohl, L.A. Peletier, Observations on blow up and dead cores for nonlinear parabolic equations, Math. Z. 202 (1989) 207–217]), the question of blow-up is investigated. Specifically, if the solution blows up, a lower bound for the time of blow-up is derived