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.
DOI:
https://doi.org/10.1090/qam/1392
Published electronically:
June 11, 2015
MathSciNet review:
3400754
Full-text PDF Free Access
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Additional Information
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, \[ \begin {array}{l} u_{t}-u_{xx}=\delta (x-vt)f\left ( u(x,t)\right ) \text { in }\Omega , \\ u(x,0)=\psi (x)\text { on }\bar {D}, \\ u(0,t)=0,u(x,t)\rightarrow 0\text { as }x\rightarrow \infty \text { for } 0<t\leq T, \end {array} \] 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.
References
- C. Y. Chan, P. Sawangtong, and T. Treeyaprasert, Existence, uniqueness and blowup for a parabolic problem with a moving nonlinear source on a semi-infinite interval, Dynam. Systems Appl. 21 (2012), no. 4, 631–643. MR 3026096
- Dean G. Duffy, Green’s functions with applications, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001. MR 1888091
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- C. M. Kirk and W. E. Olmstead, Blow-up in a reactive-diffusive medium with a moving heat source, Z. Angew. Math. Phys. 53 (2002), no. 1, 147–159. MR 1889185, DOI https://doi.org/10.1007/s00033-002-8147-6
- W. E. Olmstead, Critical speed for the avoidance of blow-up in a reactive-diffusive medium, Z. Angew. Math. Phys. 48 (1997), no. 5, 701–710. MR 1478407, DOI https://doi.org/10.1007/s000330050059
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
References
- C. Y. Chan, P. Sawangtong, and T. Treeyaprasert, Existence, uniqueness and blowup for a parabolic problem with a moving nonlinear source on a semi-infinite interval, Dynam. Systems Appl. 21 (2012), no. 4, 631–643. MR 3026096
- Dean G. Duffy, Green’s functions with applications, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001. MR 1888091 (2003e:35005)
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964. MR 0181836 (31 \#6062)
- C. M. Kirk and W. E. Olmstead, Blow-up in a reactive-diffusive medium with a moving heat source, Z. Angew. Math. Phys. 53 (2002), no. 1, 147–159. MR 1889185 (2003b:35106), DOI https://doi.org/10.1007/s00033-002-8147-6
- W. E. Olmstead, Critical speed for the avoidance of blow-up in a reactive-diffusive medium, Z. Angew. Math. Phys. 48 (1997), no. 5, 701–710. MR 1478407 (98j:35095), DOI https://doi.org/10.1007/s000330050059
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall Inc., Englewood Cliffs, N.J., 1967. MR 0219861 (36 \#2935)
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Additional Information
C. Y. Chan
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504
MR Author ID:
203257
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
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