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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A remark on the existence of suitable vector fields related to the dynamics of scalar semi-linear parabolic equations


Authors: Fengbo Hang and Huiqiang Jiang
Journal: Proc. Amer. Math. Soc. 134 (2006), 2633-2637
MSC (2000): Primary 35K20; Secondary 35B40, 54F65
Published electronically: April 7, 2006
MathSciNet review: 2213742
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Abstract | References | Similar Articles | Additional Information

Abstract: In 1992, P. Polácik showed that one could linearly imbed any vector field into a scalar semi-linear parabolic equation on $ \Omega$ with Neumann boundary condition provided that there exists a smooth vector field $ \Phi=\left( \phi_{1},\cdots,\phi_{n}\right) $ on $ \overline{\Omega}$ such that

\begin{displaymath} \left\{ \begin{array}[c]{l} \operatorname*{rank}\left( \Phi\... ...tial\nu}=0\text{ on }\partial\Omega\text{.} \end{array}\right. \end{displaymath}

In this short paper, we give a classification of all the domains on which one may find such a type of vector field.


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Additional Information

Fengbo Hang
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: fhang@math.msu.edu

Huiqiang Jiang
Affiliation: School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, Minnesota 55455
Email: hqjiang@math.umn.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08627-8
PII: S 0002-9939(06)08627-8
Received by editor(s): March 25, 2005
Published electronically: April 7, 2006
Additional Notes: The research of the first author was supported in part by NSF Grant DMS-0209504.
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.