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

Abstract

This paper studies the heat equation ut=Δu in a bounded domain Ω⊂Rn(n≥2) with positive initial data and a local nonlinear Neumann boundary condition: the normal derivative ∂u/∂n=uq on partial boundary Γ1⊆∂Ω for some q>1, while ∂u/∂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 Ω is convex and |Γ1| denotes the surface area of Γ1, then T⁎ is shown to be at least of order |Γ1|−1n−1 for n≥3 and |Γ1|−1/ln⁡(|Γ1|−1) for n=2 as |Γ1|→0, while the previous result is |Γ1|−α for any α<1n−1. Finally, we generalize the results for convex domains to the domains with only local convexity near Γ1.