Potential theory and optimal convergence rates in fast nonlinear diffusion

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A potential theoretic comparison technique is developed, which yields the conjectured optimal rate of convergence as t --> infinity for solutions of the fast diffusion equation u(t) = Delta(u(m)), (n - 2)(+)/n < m <= n/(n + 2), u, t >= 0, x is an element of R-n, n >= 1, to a spreading self-similar profile, starting from integrable initial data with sufficiently small tails. This 1/t rate is achieved uniformly in relative error, and in weaker norms such as L-1 (R-n). The range of permissible nonlinearities extends upwards towards m = 1 if the initial data shares enough of its moments with a specific self-similar profile. For example, in one space dimension, n = 1, the 1/t rate extends to the full range m is an element of]0, 1 [ of nonlinearities provided the data is correctly centered. (C) 2006 Elsevier SAS. All rights reserved.
Publisher
GAUTHIER-VILLARS/EDITIONS ELSEVIER
Issue Date
2006-07
Language
English
Article Type
Article
Keywords

POROUS-MEDIUM EQUATION; SCALAR CONSERVATION-LAWS; C-INFINITY-REGULARITY; ASYMPTOTIC-BEHAVIOR; SELF-SIMILARITY; EVOLUTION-EQUATIONS; SINGULAR DIFFUSION; WEAK SOLUTIONS; CAUCHY-PROBLEM; GAS

Citation

JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES, v.86, no.1, pp.42 - 67

ISSN
0021-7824
DOI
10.1016/j.matpur.2006.01.002
URI
http://hdl.handle.net/10203/90870
Appears in Collection
MA-Journal Papers(저널논문)
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