Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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A multi-dimensional blow-up problem due to a concentrated nonlinear source in $ \mathbb{R}^{N}$

Authors: C. Y. Chan and P. Tragoonsirisak
Journal: Quart. Appl. Math. 69 (2011), 317-330
MSC (2000): Primary 35K60, 35K57, 35B35
Published electronically: March 9, 2011
MathSciNet review: 2814530
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Abstract: Let $ B$ be an $ N$-dimensional ball $ \left\{ x\in\mathbb{R}^{N}:\left\vert x\right\vert <R\right\} $ centered at the origin with a radius $ R$, and $ \partial B$ be its boundary. Also, let $ \nu\left( x\right) $ denote the unit inward normal at $ x\in\partial B$, and let $ \chi_{B}\left( x\right) $ be the characteristic function, which is 1 for $ x\in B$, and 0 for $ x\in\mathbb{R}^{N}\setminus B$. This article studies the following multi-dimensional semilinear parabolic problem with a concentrated nonlinear source on $ \partial B$:

\begin{displaymath} \begin{array}[c]{c} u_{t}-\triangle u=\alpha{\displaystyle{\... ...\right\vert \rightarrow\infty\text{ for }0<t\leq T, \end{array}\end{displaymath}

where $ \alpha$ and $ T$ are positive numbers, $ f$ and $ \psi$ are given functions such that $ f\left( 0\right) \geq0$, $ f\left( u\right) $ and $ f^{\prime}\left( u\right) $ are positive for $ u>0$, $ f^{\prime\prime }\left( u\right) \geq0$ for $ u>0$, and $ \psi$ is nontrivial on $ \partial B,$ nonnegative, and continuous such that $ \psi\rightarrow0$ as $ \left\vert x\right\vert \rightarrow\infty,$ $ \int_{\mathbb{R}^{N}}\psi\left( x\right) dx<\infty$, and $ \triangle\psi+\alpha\left( \partial\chi_{B}\left( x\right) /\partial\nu\right) f\left( \psi\left( x\right) \right) \geq0$ in $ \mathbb{R}^{N}.$ It is shown that the problem has a unique nonnegative continuous solution before blowup occurs. We assume that $ \psi\left( x\right) =M\left( 0\right) >\psi\left( y\right) $ for $ x\in\partial B$ and $ y\notin\partial B$, where $ M\left( t\right) =\sup_{x\in\mathbb{R}^{N} }u\left( x,t\right) $. It is proved that if $ u$ blows up in a finite time, then it blows up everywhere on $ \partial B$. If, in addition, $ \psi$ is radially symmetric about the origin, then we show that if $ u$ blows up, then it blows up on $ \partial B$ only. Furthermore, if $ f\left( u\right) \geq\kappa u^{p}$, where $ \kappa$ and $ p$ are positive constants such that $ p>1$, then it is proved that for any $ \alpha$, $ u$ always blows up in a finite time for $ N\leq2$; for $ N\geq3$, it is shown that there exists a unique number $ \alpha^{\ast}$ such that $ u$ exists globally for $ \alpha\leq \alpha^{\ast}$ and blows up in a finite time for $ \alpha>\alpha^{\ast}$. A formula for computing $ \alpha^{\ast}$ is given.

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Additional Information

C. Y. Chan
Affiliation: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010
Email: chan@louisiana.edu

P. Tragoonsirisak
Affiliation: Department of Mathematics and Computer Science, Fort Valley State University, Fort Valley, Georgia 31030
Email: tragoonsirisakp@fvsu.edu

DOI: https://doi.org/10.1090/S0033-569X-2011-01219-3
Keywords: Concentrated nonlinear source, Existence, Uniqueness, Blowup.
Received by editor(s): October 12, 2009
Published electronically: March 9, 2011
Article copyright: © Copyright 2011 Brown University

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