Differential and Integral Equations

Heat equation with a nonlinear boundary condition and growing initial data

Kazuhiro Ishige and Ryuichi Sato

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

Permanent link to this document
https://projecteuclid.org/euclid.die/1493863391

Mathematical Reviews number (MathSciNet)
MR3646460

Zentralblatt MATH identifier
06738558

Subjects
Primary: 35B44: Blow-up 35K55: Nonlinear parabolic equations 35K60: Nonlinear initial value problems for linear parabolic equations

Citation

Ishige, Kazuhiro; Sato, Ryuichi. Heat equation with a nonlinear boundary condition and growing initial data. Differential Integral Equations 30 (2017), no. 7/8, 481--504. https://projecteuclid.org/euclid.die/1493863391


Export citation