A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations

Abstract

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot y(t)=g(y(t))$ on $\mathbb{R}^d$ and those of the parabolic equations $\dot u=Δu +f(x,u,\nabla u)$ on a bounded domain $Ω$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\geq 3$ and in the corresponding cases $Ω=(0,1)$, $Ω=\mathbb{T}^1$ and dim($Ω$)$\geq 2$ respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations.