BLOW-UP TIME ESTIMATES FOR A SEMILINEAR PARABOLIC PROBLEM WITH NONLINEAR MEMORY AND SMALL DIFFUSION

Abstract

We study the blow-up time of a semilinear parabolic problem with small diffusion and nonlinear memory. As the diffusion vanishes, the singular time is governed by a scalar memory equation. Under standard ellipticity, positivity and convexity assumptions, we prove convergence to this limiting time, with an  estimate in the null case and an  estimate when the potential has a positive interior maximum. Radial computations with an adaptive implicit scheme illustrate the estimates.