A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations
Résumé
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.
Mots clés
Kupka-Smale property
genericity
finite- and infinite-dimensional dynamical systems
vector fields
scalar parabolic equation
finite-and infinite-dimensional dynamical systems
vector fields
scalar
parabolic equation
Kupka-Smale property
genericity
AMS Subject Classification: 35-02
37-02
35B05
35B41
35K57
37C10
37C20
| Origine | Fichiers produits par l'(les) auteur(s) |
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