## Differential and Integral Equations

### Heat equation with a nonlinear boundary condition and growing initial data

#### Abstract

We discuss the solvability and the comparison principle for the heat equation with a nonlinear boundary condition $$\left\{ \begin{array}{ll} \partial_t u=\Delta u, & x\in\Omega,\,t > 0, \\ \nabla u\cdot\nu(x)=u^p,\qquad &x\in\partial\Omega,\,\,t > 0, \\ u(x,0)=\varphi(x)\ge 0, & x\in\Omega, \end{array} \right.$$ where $N\ge 1$, $p > 1$, $\Omega$ is a smooth domain in ${\bf R}^N$ and $\varphi(x)=O(e^{\lambda d(x)^2})$ as $d(x)\to\infty$ for some $\lambda\ge 0$. Here, $d(x)=\mbox{dist}\,(x,\partial\Omega)$. Furthermore, we obtain the lower estimates of the blow-up time of solutions with large initial data by use of the behavior of the initial data near the boundary $\partial\Omega$.

#### Article information

Source
Differential Integral Equations, Volume 30, Number 7/8 (2017), 481-504.

Dates
Accepted: November 2016
First available in Project Euclid: 4 May 2017