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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Localization for a porous medium type equation in high dimensions


Authors: Changfeng Gui and Xiaosong Kang
Journal: Trans. Amer. Math. Soc. 356 (2004), 4273-4285
MSC (2000): Primary 35K15, 35K55, 35K65; Secondary 35J40
Published electronically: May 28, 2004
MathSciNet review: 2067119
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Abstract: We prove the strict localization for a porous medium type equation with a source term, $u_{t}= \nabla(u^ {\sigma} \nabla u)+u^ \beta$, $ x \in \mathbf{R}^ N$, $ N>1$, $ \beta >\sigma +1$, $\sigma>0,$ in the case of arbitrary compactly supported initial functions $u_0$. We also otain an estimate of the size of the localization in terms of the support of the initial data $\operatorname{supp}u_0$ and the blow-up time $T$. Our results extend the well-known one dimensional result of Galaktionov and solve an open question regarding high dimensions.


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Additional Information

Changfeng Gui
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: gui@math.uconn.edu

Xiaosong Kang
Affiliation: The Fields institute, 222 College Street, Toronto, Ontario, Canada M5T 3J1
Email: xkang@fields.utoronto.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03613-X
PII: S 0002-9947(04)03613-X
Keywords: Porous medium type equation with source, localization property, blow-up, self-similar solutions, comparison
Received by editor(s): September 18, 2002
Published electronically: May 28, 2004
Article copyright: © Copyright 2004 American Mathematical Society