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  Quarterly of Applied Mathematics
Quarterly of Applied Mathematics
  
Online ISSN 1552-4485; Print ISSN 0033-569X
 

A blow-up criterion for a degenerate parabolic problem due to a concentrated nonlinear source


Authors: C. Y. Chan and R. Boonklurb
Journal: Quart. Appl. Math. 65 (2007), 781-787
MSC (2000): Primary 35K60, 35K65, 35K57
Published electronically: October 9, 2007
MathSciNet review: 2370360
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Abstract: Let $ q$, $ a$, $ b$, and $ T$ be real numbers with $ q\geq0$, $ a>0$, $ 0<b<1$, and $ T>0$. This article studies the following degenerate semilinear parabolic first initial-boundary value problem,

$\displaystyle x^{q}u_{t}(x,t)-u_{xx}(x,t)=a\delta(x-b)f\left( u(x,t)\right)$    for $\displaystyle 0<x<1,$ $\displaystyle 0<t\leq T,$    
$\displaystyle u(x,0)=\psi(x)$ for $\displaystyle 0\leq x\leq1$$\displaystyle u(0,t)=u(1,t)=0$ for $\displaystyle 0<t\leq T,$    

where $ \delta\left( x\right) $ is the Dirac delta function, and $ f$ and $ \psi$ are given functions. It is shown that for $ a$ sufficiently large, there exists a unique number $ b^{\ast}\in\left( 0,1/2\right) $ such that $ u$ never blows up for $ b\in\left( 0,b^{\ast}\right] \cup\left[ 1-b^{\ast},1\right) $, and $ u$ always blows up in a finite time for $ b\in(b^{\ast},1-b^{\ast})$. To illustrate our main results, 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
Email: chan@louisiana.edu

R. Boonklurb
Affiliation: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010
Email: rxb1828@louisiana.edu

DOI: http://dx.doi.org/10.1090/S0033-569X-07-01082-9
PII: S 0033-569X(07)01082-9
Keywords: Degenerate semilinear parabolic first initial-boundary value problem, concentrated nonlinear source, critical position, global existence, blow-up.
Received by editor(s): April 26, 2007
Published electronically: October 9, 2007
Article copyright: © Copyright 2007 Brown University



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