2008
DOI: 10.1007/s00205-008-0155-z
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Abstract: Abstract. We consider non-negative solutions of the fast diffusion equation ut = ∆u m with m ∈ (0, 1), in the Euclidean space R d , d ≥ 3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to t → ∞ for m ≥ mc = (d−2)/d, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≥ mc, or close enough to the extinction time if m < mc. Suc… Show more

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Cited by 127 publications
(319 citation statements)
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“…See also [1,25,36,39,28] for further results based on Wasserstein's distance and mass transportation theory. Concerning interpolation inequalities, weights and asymptotic behavior, we also have to quote [13,22,12] for some recent results.…”
Section: Bakry-emery Criterion For Diffusionsmentioning
confidence: 99%
See 1 more Smart Citation
“…See also [1,25,36,39,28] for further results based on Wasserstein's distance and mass transportation theory. Concerning interpolation inequalities, weights and asymptotic behavior, we also have to quote [13,22,12] for some recent results.…”
Section: Bakry-emery Criterion For Diffusionsmentioning
confidence: 99%
“…See also [1,25,36,39,28] for further results based on Wasserstein's distance and mass transportation theory. Concerning interpolation inequalities, weights and asymptotic behavior, we also have to quote [13,22,12] for some recent results.In connection with probability theory, entropy -entropy production methods have been successfully applied in a quite general framework of Riemanian manifolds: see [7,24]. However, our approach is more related to a series of attempts which have been made to establish rates of decay for solutions of diffusion equations of second and higher order: see [27,16,34].…”
mentioning
confidence: 99%
“…In the present work we establish some results in this direction by exploring in detail how 'mass' initially concentrated at spatial infinity spreads over the entire space for the super-fast diffusion equation (1.1) with m < 0. More precisely, we shall be concerned with the Cauchy problem 2) in the strongly degenerate regime m < 0, assuming the initial data v 0 ∈ C 0 (R n ) are positive, and such that v 0 (x) → +∞ as |x| → ∞ (1.3)…”
Section: Introductionmentioning
confidence: 99%
“…The exponent m * plays a very important role in the results of [2][3][4]. The proofs of convergence with rates are based on the study of the decay in time of a certain relative entropy and a careful analysis of the linearized problem which leads to certain functional inequalities of the Hardy-Poincaré type.…”
Section: Introductionmentioning
confidence: 99%