Lifespan estimates via Neumann heat kernel

This paper studies the lower bound of the lifespan $T^{\ast }$ for the heat equation $u_t=\mathrm{\Delta }u$ in a bounded domain $\Omega\subset$\mathbb{R}$^{n}(n\geq 2)$withpositiveinitialdata$u_{0}$ and a nonlinear radiation condition on partial boundary: the normal derivative $\mathrm{\partial }u\mathrm{/}\mathrm{\partial }n=u^q$ on ${\mathrm{\Gamma }}_1\subseteq \mathrm{\partial }\mathrm{\Omega }$ for some $q>1$, while $\partial u/\partial n=0$ on the other part of the boundary. Previously, under the convexity assumption of $\mathrm{\Omega }$, the asymptotic behaviors of $T^{\ast }$ on the maximum $M_0$ of $u_0$ and the surface area $\mathrm{\mid }{\mathrm{\Gamma }}_1\mathrm{\mid }$ of ${\mathrm{\Gamma }}_1$ were explored. In this paper, without the convexity requirement of $\mathrm{\Omega }$, we will show that as $M_0\to 0^{+}$, $T^{\ast }$ is at least of order $M_0^{-(q-1)}$ which is optimal. Meanwhile, we will also prove that as $\mathrm{\mid }{\mathrm{\Gamma }}_1\mathrm{\mid }\to 0^{+}$, $T^{\ast }$ is 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 $n=2$. The order on $\mathrm{\mid }{\mathrm{\Gamma }}_1\mathrm{\mid }$ when $n=2$ is almost optimal. The proofs are carried out by analyzing the representation formula of $u$ in terms of the Neumann heat kernel.