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 $\dot{u}=\Delta u+f(x,u,\nabla u)$ on a bounded domain $\Omega$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\ge 3$ and in the corresponding cases $\Omega =(0,1)$, $\Omega ={\mathbb{T}}^1$ and dim($\Omega$)$\ge 2$ 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.

References in corpus (1)

• Generic Morse-Smale property for the parabolic equation on the circle

Cited by in corpus (1)

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