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

Authors:	Fengbo Hang Huiqiang Jiang

Institution:	Department of Mathematics, Michigan State University, East Lansing, Michigan 48824 ; School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, Minnesota 55455

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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