Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods

Matteo Bonforte; Jean Dolbeault; Matteo Muratori; Bruno Nazaret

doi:10.3934/krm.2017003

Article Contents

2017, Volume 10, Issue 1: 61-91. Doi: 10.3934/krm.2017003

This issue Previous Article Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Next Article A kinetic games framework for insurance plans

Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods

1.
Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain
2.
Ceremade, UMR CNRS $n^{\circ }$ 7534, Université Paris-Dauphine, PSL Research University, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France
3.
Dipartimento di Matematica Felice Casorati, Università degli Studi di Pavia, Via A. Ferrata 5,27100 Pavia, Italy
4.
SAMM, Université Paris 1, 90, rue de Tolbiac, 75634 Paris Cedex 13, France

* Corresponding author
* Corresponding author

Received: February 2016

Revised: June 2016

Published: September 2017

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Abstract

This paper is the second part of the study. In Part Ⅰ, self-similar solutions of a weighted fast diffusion equation (WFD) were related to optimal functions in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities (CKN) applied to radially symmetric functions. For these inequalities, the linear instability (symmetry breaking) of the optimal radial solutions relies on the spectral properties of the linearized evolution operator. Symmetry breaking in (CKN) was also related to large-time asymptotics of (WFD), at formal level. A first purposPart Ⅱis to give a rigorous justification of this point, that is, to determine the asymptotic rates of convergence of the solutions to (WFD) in the symmetry range of (CKN) as well as in the symmetry breaking range, and even in regimes beyond the supercritical exponent in (CKN). Global rates of convergence with respect to a free energy (or entropy) functional are also investigated, as well as uniform convergence to self-similar solutions in the strong sense of the relative error. Differences with large-time asymptotics of fast diffusion equations without weights are emphasized.

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Mathematics Subject Classification: Primary: 35K55, 35B40, 49K30; Secondary: 26D10, 35B06, 46E35, 49K20, 35J20.

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Figure 1. The spectrum of $\mathcal L$ as a function of $\delta=\frac1{1-m}$, with $n=5$. The essential spectrum corresponds to the grey area, and its bottom is determined by the parabola $\delta\mapsto\Lambda_{\rm ess}(\delta)$. The two eigenvalues $\Lambda_{0,1}$ and $\Lambda_{1,0}$ are given by the plain, half-lines, away from the essential spectrum. Note that solutions of the eigenvalue problem exist for any value of $\delta$ but may not be in the domain of the operator or below the essential spectrum and are then represented as dotted half-lines. See [4,Appendix B] for a discussion of the values of $\delta_1$, $\delta_2$, ... $\delta_5$. The right figure is an enlargement of the left one. This configuration is not generic: see [4,Fig. 5] for other cases.

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Figure 2. The intrinsic cylinders $Q_r(t_0,x_0)$, $Q^+_r(t_0,x_0)$ and $Q^-_r(t_0,x_0)$.

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