Title: 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
$\Gamma_{1}$.