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

Abstract

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