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Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Blow-up criteria for a parabolic problem due to a concentrated nonlinear source on a semi-infinite interval

Authors: C. Y. Chan and T. Treeyaprasert
Journal: Quart. Appl. Math. 70 (2012), 159-169
MSC (2010): Primary 35K60, 35K57
Published electronically: August 30, 2011
MathSciNet review: 2920621
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Abstract: Let $ \alpha$, $ b$ and $ T$ be positive numbers, $ D=\left( 0,\infty\right)$, $ \bar{D}=\left[0,\infty\right)$, and $ \Omega=D\times\left(0,T\right]$. This article studies the first initial-boundary value problem,

\begin{displaymath} \begin{array}[c]{c} u_{t}-u_{xx}=\alpha\delta(x-b)f\left( u(... ...im_{x\rightarrow\infty}u(x,t)\text{ for }0<t\leq T, \end{array}\end{displaymath}

where $ \delta\left( x\right) $ is the Dirac delta function, and $ f$ and $ \psi$ are given functions. We assume that $ f\left( 0\right) \geq0$, $ f(u)$ and its derivatives $ f^{\prime}(u)$ and $ f^{\prime\prime}\left( u\right) $ are positive for $ u>0$, and $ \psi(x)$ is nontrivial, nonnegative and continuous such that $ \psi\left( 0\right) =0=\lim_{x\rightarrow\infty}\psi\left( x\right) $, and

$\displaystyle \psi^{\prime\prime}+\alpha\delta(x-b)f\left( \psi\right) \geq0$ in $\displaystyle D$.

It is shown that if $ u$ blows up, then it blows up in a finite time at the single point $ b$ only. A criterion for $ u$ to blow up in a finite time and a criterion for $ u$ to exist globally are given. It is also shown that there exists a critical position $ b^{\ast}$ for the nonlinear source to be placed such that no blowup occurs for $ b\leq b^{\ast}$, and $ u$ blows up in a finite time for $ b>b^{\ast}$. This also implies that $ u$ does not blow up in infinite time. The formula for computing $ b^{\ast}$ is also derived. For illustrations, two examples are given.

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Additional Information

C. Y. Chan
Affiliation: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010

T. Treeyaprasert
Affiliation: Department of Mathematics and Statistics, Thammasat University Rangsit Center, Pathumthani, 12121 Thailand

Keywords: Semilinear parabolic first initial-boundary value problem, semi-infinite interval, concentrated nonlinear source, single blow-up point, critical position, global existence
Received by editor(s): August 26, 2010
Published electronically: August 30, 2011
Article copyright: © Copyright 2011 Brown University

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