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 = 𝕋^{1} 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.

Romain Joly ^{1};
Geneviève Raugel ^{1}

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Romain Joly; Geneviève Raugel. A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations. Confluentes Mathematici, Volume 3 (2011) no. 3, pp. 471-493. doi : 10.1142/S1793744211000369. https://cml.centre-mersenne.org/articles/10.1142/S1793744211000369/

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