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July/August 2017 Heat equation with a nonlinear boundary condition and growing initial data
Kazuhiro Ishige, Ryuichi Sato
Differential Integral Equations 30(7/8): 481-504 (July/August 2017).

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 $\partial\Omega$.

Citation

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Kazuhiro Ishige. Ryuichi Sato. "Heat equation with a nonlinear boundary condition and growing initial data." Differential Integral Equations 30 (7/8) 481 - 504, July/August 2017.

Information

Accepted: 1 November 2016; Published: July/August 2017
First available in Project Euclid: 4 May 2017

zbMATH: 06738558
MathSciNet: MR3646460

Subjects:
Primary: 35B44, 35K55, 35K60

Rights: Copyright © 2017 Khayyam Publishing, Inc.

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Vol.30 • No. 7/8 • July/August 2017
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