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

 

On the blow-up of heat flow for conformal $3$-harmonic maps


Authors: Chao-Nien Chen, L. F. Cheung, Y. S. Choi and C. K. Law
Journal: Trans. Amer. Math. Soc. 354 (2002), 5087-5110
MSC (2000): Primary 35K50, 35K65, 58E20
Published electronically: July 16, 2002
MathSciNet review: 1926851
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Abstract | References | Similar Articles | Additional Information

Abstract: Using a comparison theorem, Chang, Ding, and Ye (1992) proved a finite time derivative blow-up for the heat flow of harmonic maps from $D^2$ (a unit ball in ${\mathbf R}^2$) to $S^2$ (a unit sphere in ${\mathbf R}^3$) under certain initial and boundary conditions. We generalize this result to the case of $3$-harmonic map heat flow from $D^3$ to $S^3$. In contrast to the previous case, our governing parabolic equation is quasilinear and degenerate. Technical issues such as the development of a new comparison theorem have to be resolved.


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Additional Information

Chao-Nien Chen
Affiliation: Department of Mathematics, National Changhua University of Education, Changhua, Taiwan, Republic of China
Email: chenc@math.ncue.edu.tw

L. F. Cheung
Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
Email: matheclf@maun01.ma.polyu.edu.hk

Y. S. Choi
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: choi@math.uconn.edu

C. K. Law
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 80424, Republic of China
Email: law@math.nsysu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03054-4
PII: S 0002-9947(02)03054-4
Keywords: $p$-harmonic maps, heat flow, blow up, maximum principle
Received by editor(s): June 25, 2001
Received by editor(s) in revised form: December 21, 2001
Published electronically: July 16, 2002
Dedicated: Dedicated to Shui-Nee Chow on the occasion of his 60th birthday
Article copyright: © Copyright 2002 American Mathematical Society