Generalized Sobolev Inequalities and Asymptotic Behaviour in Fast Diffusion and Porous Medium Problems
Dolbeault, Jean; Del Pino, Manuel (1999), Generalized Sobolev Inequalities and Asymptotic Behaviour in Fast Diffusion and Porous Medium Problems. https://basepub.dauphine.fr/handle/123456789/6432
TypeDocument de travail / Working paper
Series titleCahiers du CEREMADE
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Abstract (EN)In this paper we prove a new family of inequalities which is intermediate between the classical Sobolev inequalities and the Gross logarithmic Sobolev inequality by the minimization of a well choosen functional and the use of recent uniqueness results for the ground state of the corresponding nonlinear scalar field equation, which allows us to identify the optimal constants . This result is then applied to the equation ut = ∆u m in IR for m ∈ [ N , 1[ (fast diffusion) and m > 1 (porous medium), thus giving an exponential rate of decay for the relative entropy to the stationary solution of a rescaled problem and describing the intermediate asymptotics in the L1(IR )-norm.
Subjects / KeywordsSobolev embeddings; Optimal constants; Minimization; Radial symmetry; Uniqueness; Fast diffusion; Time-dependent rescaling; Porous medium; Relative entropy; Optimal rate of decay; Intermediate asymptotics
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