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
TypeArticle accepté pour publication ou publié
Journal nameJournal of Mathematical Analysis and Applications
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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 / KeywordsOptimal 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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Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the p-Laplacian Del Pino, Manuel; Dolbeault, Jean (2002) Article accepté pour publication ou publié