Title: Lifespan Estimate for the Nonlinear Radiation on Partial Boundary.

Speaker: Zhengfang Zhou, Professor, Michigan State University

Place: Math.Building 621Room

Time: 15:00-16:00 June 22 2018.

Abstract: This talk studies the upper and lower bounds of the lifespan $T^{*}$ for the heat equation $u_t=\Delta u$ in a bounded domain $\Omega\subset\mathbb{R}^{n}(n\geq 2)$ with positive initial data $u_{0}$ and a partial radiation 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 of the boundary. Previously, under the convexity assumption of $\Omega$, the asymptotic behaviors of $T^{*}$ on the maximum $M_{0}$ of $u_{0}$ and the surface area $|\Gamma_{1}|$ of $\Gamma_{1}$ were explored. In this talk, we will show that as $M_{0}\rightarrow 0^{+}$, $T^{*}$ is at least of order $M_{0}^{-(q-1)}$ which is optimal. On the other hand, as $|\Gamma_{1}|\rightarrow 0^{+}$, $T^{*}$ will be 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 $n=2$. The order on $|\Gamma_{1}|$ when $n=2$ is almost optimal. The improvement is obtained by carefully analyzing the representation formula of $u$ in terms of the Neumann heat kernel, these lower bounds for $T^{*}$ don’t require the convexity of $\Omega$.