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

Abstract

This paper studies the heat equation \(u_t=\Delta u\) in a bounded domain \(\Omega\subset\mathbb{R}^{n}(n\geq 2)\) with positive initial data and a local nonlinear Neumann boundary condition: the normal derivative \(\partial u/\partial n=u^{q}\) on partial boundary \(\Gamma_1\subseteq \partial\Omega\) for some \(q>1\), while \(\partial u/\partial n=0\) 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 \(q\rightarrow 1^{+}\). Since the existing upper bound is of order \((q-1)^{-1}\), this result is sharp. Secondly, if \(\Omega\) is convex and \(|\Gamma_{1}|\) denotes the surface area of \(\Gamma_{1}\), then \(T^{*}\) is shown to be at least of order \(|\Gamma_{1}|^{-\frac{1}{n-1}}\) for \(n\geq 3\) and \(|\Gamma_{1}|^{-1}\big/\ln\big(|\Gamma_{1}|^{-1}\big)\) for \(n=2\) as \(|\Gamma_{1}|\rightarrow 0\), while the previous result is \(|\Gamma_{1}|^{-\alpha}\) for any \(\alpha<\frac{1}{n-1}\). Finally, we generalize the results for convex domains to the domains with only local convexity near \(\Gamma_{1}\).