Nonlinear diffusions, hypercontractivity and the optimal Lp-Euclidean logarithmic Sobolev inequality

Del Pino, Manuel; Dolbeault, Jean; Gentil, Ivan

Del Pino, Manuel; Dolbeault, Jean; Gentil, Ivan (2004), Nonlinear diffusions, hypercontractivity and the optimal Lp-Euclidean logarithmic Sobolev inequality, Journal of Mathematical Analysis and Applications, 293, 2, p. 375-388. http://dx.doi.org/10.1016/j.jmaa.2003.10.009

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Type

Article accepté pour publication ou publié

Date

2004

Nom de la revue

Journal of Mathematical Analysis and Applications

Volume

293

Numéro

Éditeur

Elsevier

Pages

375-388

Résumé (EN)

The equation ut=Δp(u1/(p−1)) for p>1 is a nonlinear generalization of the heat equation which is also homogeneous, of degree 1. For large time asymptotics, its links with the optimal Lp-Euclidean logarithmic Sobolev inequality have recently been investigated. Here we focus on the existence and the uniqueness of the solutions to the Cauchy problem and on the regularization properties (hypercontractivity and ultracontractivity) of the equation using the Lp-Euclidean logarithmic Sobolev inequality. A large deviation result based on a Hamilton–Jacobi equation and also related to the Lp-Euclidean logarithmic Sobolev inequality is then stated.

Mots-clés

Optimal Lp-Euclidean logarithmic Sobolev inequality; Sobolev inequality; Nonlinear parabolic equations; Degenerate parabolic problems; Entropy; Existence; Cauchy problem; Uniqueness; Regularization; Hypercontractivity; Ultracontractivity; Large deviations; Hamilton–Jacobi equations