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

Abstract

In 1992, P. Poláčik\cite{P2} showed that one could linearly imbed any vector fields into a scalar semi-linear parabolic equation on $Ω$ with Neumann boundary condition provided that there exists a smooth vector field $Φ=(ϕ_{1},...,ϕ_{n}) $ on $\barΩ$ such that \[ \left\{\begin{array} [c]{l}% \operatorname*{rank}(Φ(x) ,\partial_{1}Φ(x) ,...,\partial_{n}Φ(x)) =n\text{for all}x\in\barΩ, \frac{\partialΦ}{\partialν}=0\text{on}\partialΩ\text{.}% \end{array} \right. \] In this short note, we give a classification of all the domains on which one may find such type of vector fields.