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

This paper studies the heat equation $u_t=\mathrm{\Delta }u$ in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 2)$ with positive initial data and a local nonlinear Neumann boundary condition: the normal derivative $\partial u/\partial n=u^{q}$onpartialboundary$\Gamma_1\subseteq \partial\Omega$ for some $q>1$, while $\mathrm{\partial }u\mathrm{/}\mathrm{\partial }n=0$ on the other part. We investigate the lower bound of the blow-up time $T^{\ast }$ of $u$ in several aspects. First, $T^{\ast }$ is proved to be at least of order $(q-1{)}^{-1}$ as $q\to 1^{+}$. Since the existing upper bound is of order $(q-1{)}^{-1}$, this result is sharp. Secondly, if $\mathrm{\Omega }$ is convex and $\mathrm{\mid }{\mathrm{\Gamma }}_1\mathrm{\mid }$ denotes the surface area of ${\mathrm{\Gamma }}_1$, then $T^{\ast }$ is shown to be at least of order $\mathrm{\mid }{\mathrm{\Gamma }}_1{\mathrm{\mid }}^{-\frac{1}{n-1}}$ for $n\ge 3$ and $\mathrm{\mid }{\mathrm{\Gamma }}_1{\mathrm{\mid }}^{-1}/\ln \left(\mathrm{\mid }{\mathrm{\Gamma }}_1{\mathrm{\mid }}^{-1}\right)$ for $n=2$ as $\mathrm{\mid }{\mathrm{\Gamma }}_1\mathrm{\mid }\to 0$, while the previous result is $\mathrm{\mid }{\mathrm{\Gamma }}_1{\mathrm{\mid }}^{-\alpha }$ for any $\alpha <\frac{1}{n-1}$. Finally, we generalize the results for convex domains to the domains with only local convexity near ${\mathrm{\Gamma }}_1$.