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

Hang, Fengbo; Jiang, Huiqiang

doi:10.1090/S0002-9939-06-08627-8

A remark on the existence of suitable vector fields related to the dynamics of scalar semi-linear parabolic equations
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by Fengbo Hang and Huiqiang Jiang PDF

Proc. Amer. Math. Soc. 134 (2006), 2633-2637 Request permission

Abstract:

In 1992, P. Poláčik 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 \[ \left \{ \begin {array} [c]{l} \operatorname {rank}\left ( \Phi \left ( x\right ) ,\partial _{1}\Phi \left ( x\right ) ,\cdots ,\partial _{n}\Phi \left ( x\right ) \right ) =n\text { for all }x\in \overline {\Omega }, \frac {\partial \Phi }{\partial \nu }=0\text { on }\partial \Omega \text {.} \end {array} \right . \] 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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