Lower bounds for the blow-up time of the heat equation in convex domains with local nonlinear boundary conditions

Abstract

This paper studies the lower bound for the blow-up time $T^{*}$ of the heat equation $u_t=\Delta u$ in a bounded convex domain $\Omega$ in $\mathbb{R}^{N}(N\geq 2)$ with positive initial data $u_{0}$ 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. For any $\alpha<\frac{1}{N-1}$, we obtain a lower bound for $T^{*}$ which is of order $|\Gamma_{1}|^{-\alpha}$ as $|\Gamma_{1}|\rightarrow 0^{+}$, where $|\Gamma_{1}|$ represents the surface area of $\Gamma_{1}$. As $|\Gamma_{1}|\rightarrow 0^{+}$, this result significantly improves the previous lower bound $\ln\big(|\Gamma_1|^{-1}\big)$ and is almost optimal in dimension $N=2$, since the existing upper bound is of order $|\Gamma_{1}|^{-1}$ as $|\Gamma_{1}|\rightarrow 0^{+}$. In addition, the optimal asymptotic order of the lower bound for $T^{*}$ on $q$ (as $q\rightarrow 1^{+}$) and on $M_{0}$ (as $M_{0}\rightarrow 0^{+}$) are obtained, where $M_{0}$ denotes the maximum of $u_{0}$.