Differential and Integral Equations

Heat equation with a nonlinear boundary condition and growing initial data

Kazuhiro Ishige and Ryuichi Sato

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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

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

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

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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