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

Abstract

This paper studies the heat equation $u_t=Δu$ in a bounded domain $Ω\subset\mathbb{R}^{n}(n\geq 2)$ with positive initial data and a local nonlinear Neumann boundary condition: the normal derivative $\partial u/\partial n=u^{q}$ on partial boundary $Γ_1\subseteq \partialΩ$ 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 $Ω$ is convex and $|Γ_{1}|$ denotes the surface area of $Γ_{1}$, then $T^{*}$ is shown to be at least of order $|Γ_{1}|^{-\frac{1}{n-1}}$ for $n\geq 3$ and $|Γ_{1}|^{-1}\big/\ln\big(|Γ_{1}|^{-1}\big)$ for $n=2$ as $|Γ_{1}|\rightarrow 0$, while the previous result is $|Γ_{1}|^{-α}$ for any $α<\frac{1}{n-1}$. Finally, we generalize the results for convex domains to the domains with only local convexity near $Γ_{1}$.