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

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
Journal name
Journal of Mathematical Analysis and Applications
Volume
293
Number
2
Publisher
Elsevier
Pages
375-388
Publication identifier
http://dx.doi.org/10.1016/j.jmaa.2003.10.009
Metadata
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Author(s)
Del Pino, Manuel
Dolbeault, Jean cc
Gentil, Ivan
Abstract (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.
Subjects / Keywords
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

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