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

Joly, Romain; Raugel, Geneviève

Confluentes Mathematici, Tome 3 (2011) no. 3, pp. 471-493.

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations ẏ(t) = g(y(t)) on ℝ^d and those of the parabolic equations $\dot{u}=\mathrm{\Delta }u+f(x,u,\mathrm{\nabla }u)$ on a bounded domain Ω. We give details on the similarities of these dynamics in the cases d = 1, d = 2 and d ≥ 3 and in the corresponding cases Ω = (0, 1), Ω = 𝕋¹ and dim(Ω) ≥ 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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