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

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

2010-05-11

arxiv.org/abs/1005.1727v2

Mathematics

Analysis of PDEs

Scientific paper

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot y(t)=g(y(t))$ on $\Rm^d$ and those of the parabolic equations $\dot u=\Delta u +f(x,u,\grad u)$ on a bounded domain $\Omega$. 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 $\Omega=(0,1)$, $\Omega=\Tm^1$ and dim($\Omega$)$\geq 2$ 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.

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