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 $T^{\ast }$ of the heat equation $u_t=\Delta u$ in a bounded convex domain $\Omega$ in ${\mathbb{R}}^N(N\ge 2)$ 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 $T^{\ast }$ which is of order $|{\Gamma }_1{|}^{-\alpha }$ as $|{\Gamma }_1|\to 0^{+}$, where $|{\Gamma }_1|$ represents the surface area of ${\Gamma }_1$. As $|{\Gamma }_1|\to 0^{+}$, this result significantly improves the previous lower bound $\ln (|{\Gamma }_1{|}^{-1})$ and is almost optimal in dimension $N=2$, since the existing upper bound is of order $|{\Gamma }_1{|}^{-1}$ as $|{\Gamma }_1|\to 0^{+}$. In addition, the optimal asymptotic order of the lower bound for $T^{\ast }$ on $q$ (as $q\to 1^{+}$) and on $M_0$ (as $M_0\to 0^{+}$) are obtained, where $M_0$ denotes the maximum of $u_0$.

21 pages

References in corpus (1)

• Blow-up problems for the heat equation with a local nonlinear Neumann boundary condition

Cited by in corpus (1)

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