Improvements on lower bounds for the blow-up time under local nonlinear Neumann conditions

This paper studies the heat equation u(t) = Delta u in a bounded domain Omega subset of R-n (n >= 2) with positive initial data and a local nonlinear Neumann boundary condition: the normal derivative partial derivative u/partial derivative n = u(q) on partial boundary Gamma(1) subset of partial derivative Omega for some q > 1, while partial derivative u/partial derivative n = 0 on the other part. We investigate the lower bound of the blow-up time T* of u in several aspects. First, T* is proved to be at least of order (q - 1)(-1) as q -> 1(+). Since the existing upper bound is of order (q - 1)(-1). this result is sharp. Secondly, if Omega is convex and vertical bar Gamma(1)vertical bar denotes the surface area of Gamma(1), then T * is shown to be at least of order vertical bar Gamma(1)vertical bar(-1/n-1) for n >= 3 and vertical bar Gamma(1)vertical bar(-1)/ln(vertical bar Gamma(1)vertical bar(-1)) for n = 2 as vertical bar Gamma(1)vertical bar -> 0, while the previous result is vertical bar Gamma(1)vertical bar(-alpha) for any alpha < 1/n-1. Finally, we generalize the results for convex domains to the domains with only local convexity near Gamma(1). (C) 2018 Elsevier Inc. All rights reserved.