Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Single blow-up point and critical speed for a parabolic problem with a moving nonlinear source on a semi-infinite interval

Authors: C. Y. Chan, P. Sawangtong and T. Treeyaprasert
Journal: Quart. Appl. Math. 73 (2015), 483-492
MSC (2010): Primary 35K60, 35B35, 35K55, 35K57
Published electronically: June 11, 2015
MathSciNet review: 3400754
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Abstract: Let $ v$ and $ T$ be positive numbers, $ D=\left ( 0,\infty \right ) $, $ \Omega =D\times \left ( 0,T\right ] $, and $ \bar {D}$ be the closure of $ D$. This article studies the first initial-boundary value problem,

$\displaystyle \begin {array}{l} u_{t}-u_{xx}=\delta (x-vt)f\left ( u(x,t)\right... ...ghtarrow 0\text { as }x\rightarrow \infty \text { for } 0<t\leq T, \end{array} $

where $ \delta \left ( x\right ) $ is the Dirac delta function, and $ f$ and $ \psi $ are given functions. It is shown that if the solution $ u$ blows up in a finite time $ t_{b}$, then it blows up only at the point $ x=vt_{b}$. A criterion for $ u$ to exist globally and a criterion for $ u$ to blow up in a finite time are given. Furthermore, the problem is shown to have a critical speed $ v^{\ast }$ of the moving nonlinear source such that no blowup occurs for $ v\geq v^{\ast }$ and blowup occurs in a finite time for $ v<v^{\ast }$. The formula for computing $ v^{\ast }$ is also derived.

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Additional Information

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

P. Sawangtong
Affiliation: Department of Mathematics, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
Email: panumarts@kmutnb.ac.th

T. Treeyaprasert
Affiliation: Department of Mathematics and Statistics, Thammasat University, Rangsit Center, Pathumthani 12120, Thailand
Email: tawikan@tu.ac.th

DOI: https://doi.org/10.1090/qam/1392
Keywords: Semilinear parabolic first initial-boundary value problem, single blow-up point, critical speed of the moving nonlinear source.
Received by editor(s): September 3, 2013
Published electronically: June 11, 2015
Article copyright: © Copyright 2015 Brown University

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