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

Abstract

In 1992, P. Poláčik showed that one could linearly imbed any vector field into a scalar semi-linear parabolic equation on Ω \Omega with Neumann boundary condition provided that there exists a smooth vector field Φ = ( ϕ 1 , ⋯ , ϕ n ) \Phi =\left ( \phi _{1},\cdots ,\phi _{n}\right ) on Ω ¯ \overline {\Omega } such that \[ { rank ⁡ ( Φ ( x ) , ∂ 1 Φ ( x ) , ⋯ , ∂ n Φ ( x ) ) = n for all x ∈ Ω ¯ , ∂ Φ ∂ ν = 0 on ∂ Ω . \left \{ \begin {array} [c]{l} \operatorname {rank}\left ( \Phi \left ( x\right ) ,\partial _{1}\Phi \left ( x\right ) ,\cdots ,\partial _{n}\Phi \left ( x\right ) \right ) =n\text { for all }x\in \overline {\Omega }, \frac {\partial \Phi }{\partial \nu }=0\text { on }\partial \Omega \text {.} \end {array} \right . \] In this short paper, we give a classification of all the domains on which one may find such a type of vector field.