Asymptotics of the Fast Diffusion Equation via Entropy Estimates

By Adrien Blanchet, Matteo Bonforte, Jean Dolbeault, G. Grillo, Juan Luis V'azquez. 2007

169 citations recorded by the source corpus.

We consider non-negative solutions of the fast diffusion equation ut = Δum with m ∈ (0, 1) in the Euclidean space $${{\mathbb R}^d}$$, 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. Such results are new in the range m ≦ mc where previous approaches fail. In the range mc < m < 1, we improve on known results.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/98993a5edc12d4c6937561f9448d293edc9376ab

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Citations in the corpus

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, methodology): “We refer to [171] and [242] for the earlier extinction analysis for so-called small solutions, and to [39], [36] for stabilization of solution with certain fat tails to the self-similar solutions called pseudoBarenblatt solutions.”
Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry (2010, unknown): “We may replace the power by an exponential yielding the notion of "extended sub-exponential law", i.e. given any convex function V : Rn → (0,∞) and p > 0, we shall say that dµ(x) = e−V(x) p dx is an "extended sub-exponential like law".”
Stability Results for Logarithmic Sobolev and Gagliardo–Nirenberg Inequalities (2014, methodology): “Closer to our approach is Bobkov et al. [2014], Fathi et al. [2014], who exploit the difference between the inequality of [Stam, 1959, Inequality (2.”
Weighted poincaré-type inequalities for cauchy and other convex measures (2009, background): “After this work was completed, we became aware of the related references [2, 3], where some similar inequalities are proved and analyzed by different methods and with different purposes.”
A fractional Yamabe flow and some applications (2011, background): “In the classical case σ = 1, the extinction profiles of solutions of porous medium equations have been described in the results of [34, 29, 26, 7, 8] and so on.”