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12 results on '"Hardy-Poincaré inequalities"'

1. Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations.

Author

Bonforte, Matteo, Dolbeault, Jean, Nazaret, Bruno, and Simonov, Nikita

Subjects
HEAT equation, EVOLUTION equations, INTERPOLATION, FAMILY stability, REACTION-diffusion equations, ENTROPY
Abstract

We provide a scheme of a recent stability result for a family of Gagliardo-Nirenberg-Sobolev (GNS) inequalities, which is equivalent to an improved entropy – entropy production inequality associated with an appropriate fast diffusion equation (FDE) written in self-similar variables. This result can be rephrased as an improved decay rate of the entropy of the solution of (FDE) for well prepared initial data. There is a family of Caffarelli-Kohn-Nirenberg (CKN) inequalities which has a very similar structure. When the exponents are in a range for which the optimal functions for (CKN) are radially symmetric, we investigate how the methods for (GNS) can be extended to (CKN). In particular, we prove that the solutions of the evolution equation associated to (CKN) also satisfy an improved decay rate of the entropy, after an explicit delay. However, the improved rate is obtained without assuming that initial data are well prepared, which is a major difference with the (GNS) case. [ABSTRACT FROM AUTHOR]

Published
2023
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3. Inverse problem for a degenerate/singular parabolic system with Neumann boundary conditions.

Author

Alaoui, Mohammed, Hajjaj, Abdelkarim, Maniar, Lahcen, and Salhi, Jawad

Abstract

In this paper, we study an inverse source problem for a degenerate and singular parabolic system where the boundary conditions are of Neumann type. We consider a problem with degenerate diffusion coefficients and singular lower-order terms, both vanishing at an interior point of the space domain. In particular, we address the question of well-posedness of the problem, and then we prove a stability estimate of Lipschitz type in determining the source term by data of only one component. Our method is based on Carleman estimates, cut-off procedures and a reflection technique. [ABSTRACT FROM AUTHOR]

Published
2021
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4. Functional Inequalities: Nonlinear Flows and Entropy Methods as a Tool for Obtaining Sharp and Constructive Results.

Author

Dolbeault, Jean

Abstract

Interpolation inequalities play an essential role in analysis with fundamental consequences in mathematical physics, nonlinear partial differential equations (PDEs), Markov processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as an optimal rate of decay of an entropy for an associated evolution equation. Much more has been learned by adopting this point of view. This paper aims at illustrating some of these recent aspect of entropy-entropy production inequalities, with applications to stability in Gagliardo–Nirenberg–Sobolev inequalities and symmetry results in Caffarelli–Kohn–Nirenberg inequalities. Entropy methods provide a framework which relates nonlinear regimes with their linearized counterparts. This framework allows to prove optimality results, symmetry results and stability estimates. Some emphasis will be put on the hidden structure which explain such properties. Related open problems will be listed. [ABSTRACT FROM AUTHOR]

Published
2021
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5. Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method

Author

Bonforte, Matteo, Dolbeault, Jean, Nazaret, Bruno, Simonov, Nikita, Instituto de Ciencias Matemàticas [Madrid] (ICMAT), Universidad Carlos III de Madrid [Madrid] (UC3M)-Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM)-Universidad Autónoma de Madrid (UAM)-Consejo Superior de Investigaciones Científicas [Madrid] (CSIC), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Université Paris 1 Panthéon-Sorbonne (UP1), Fédération Parisienne de Modélisation Mathématique (FP2M), Centre National de la Recherche Scientifique (CNRS), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Project MTM2017-85757-P (Ministry of Science and Innovation, Spain) and the Spanish Ministry of Science and Innovation, through the 'Severo Ochoa Programme for Centres of Excellence in R&D' (CEX2019-000904-S)E.U. H2020 MSCA programme, grant agreement 777822Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR)Inria Mokaplan teamDIM Math-Innov of the Region Île-de-France, ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), and European Project: 777822,GHAIA(2017)

Subjects
intermediate asymptotics, Mathematics::Analysis of PDEs, fast diffusion equation, rates of convergence, stability, Harnack Principle, Hardy-Poincaré inequalities, spectral gap, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], self-similar Barenblatt solutions, asymptotic behavior, 2020 Mathematics Subject Classification.26D10, 46E35, 35K55, 49J40, 35B40, 49K20, 49K30, 35J20, Gagliardo-Nirenberg inequality, entropy methods
Abstract

Ce document contient hal-02887010, v1: Stability in Gagliardo-Nirenberg inequalities and hal-02887013, v1: Stability in Gagliardo-Nirenberg inequalities. Supplementary material, avec plusieurs additions dont: Chapitre 1 (variational methods) et Chapitre 6 (Sobolev's inequality).; International audience; The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard Sobolev inequality (in dimension larger than three), or Onofri's inequality in dimension two. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. The regularity properties of the parabolic flow allow us to connect an improved entropy - entropy production inequality during an initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymptotic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms (a Fisher information which can be interpreted as a generalized Heisenberg uncertainty principle) the distance to the manifold of optimal functions. The method is constructive and, for the first time, quantitative estimates of the stability constant are obtained, including in the critical case of Sobolev's inequality. To build the estimates, we establish a quantitative global Harnack principle and perform a detailed analysis of large time asymptotics by entropy methods.

Published
2022

6. Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations

Author

Bonforte, Matteo, Dolbeault, Jean, Nazaret, Bruno, Simonov, Nikita, Universidad Autónoma de Madrid (UAM), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Université Paris 1 Panthéon-Sorbonne (UP1), Université d'Évry-Val-d'Essonne (UEVE), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), Université Paris sciences et lettres (PSL), and UAM. Departamento de Matemáticas

Subjects
Self-Similar Solutions, Matemáticas, Applied Mathematics, Caffarelli-Kohn-Nirenberg inequality, selfsimilar solutions, fast diffusion equation, rates of convergence, stability, Harnack Principle, Mathematics - Analysis of PDEs, Hardy-Poincaré inequalities, spectral gap, FOS: Mathematics, Gagliardo-Nirenberg-Sobolev inequality, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Discrete Mathematics and Combinatorics, 26D10, 46E35, 35K55, 35B40, 49K20, Analysis, entropy methods, Analysis of PDEs (math.AP)
Abstract

International audience; We provide a scheme of a recent stability result for a family of Gagliardo-Nirenberg-Sobolev (GNS) inequalities, which is equivalent to an improved entropy-entropy production inequality associated with an appropriate fast diffusion equation (FDE) written in self-similar variables. This result can be rephrased as an improved decay rate of the entropy of the solution of (FDE) for well prepared initial data. There is a family of Caffarelli-Kohn-Nirenberg (CKN) inequalities which has a very similar structure. When the exponents are in a range for which the optimal functions for (CKN) are radially symmetric, we investigate how the methods for (GNS) can be extended to (CKN). In particular, we prove that the solutions of the evolution equation associated to (CKN) also satisfy an improved decay rate of the entropy, after an explicit delay. However, the improved rate is obtained without assuming that initial data are well prepared, which is a major difference with the (GNS) case.

Published
2022
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7. Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method

Author

Bonforte, Matteo, Dolbeault, Jean, Nazaret, Bruno, Simonov, Nikita, Instituto de Ciencias Matemàticas [Madrid] (ICMAT), Universidad Autónoma de Madrid (UAM)-Consejo Superior de Investigaciones Científicas [Madrid] (CSIC)-Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM)-Universidad Carlos III de Madrid [Madrid] (UC3M), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Université Paris 1 Panthéon-Sorbonne (UP1), Fédération Parisienne de Modélisation Mathématique (FP2M), Centre National de la Recherche Scientifique (CNRS), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Project MTM2017-85757-P (Ministry of Science and Innovation, Spain) and the Spanish Ministry of Science and Innovation, through the 'Severo Ochoa Programme for Centres of Excellence in R&D' (CEX2019-000904-S)E.U. H2020 MSCA programme, grant agreement 777822Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR)Inria Mokaplan teamDIM Math-Innov of the Region Île-de-France, ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), European Project: 777822,GHAIA(2017), Universidad Autonoma de Madrid (UAM)-Consejo Superior de Investigaciones Científicas [Madrid] (CSIC)-Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM)-Universidad Carlos III de Madrid [Madrid] (UC3M), Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)

Subjects
intermediate asymptotics, Mathematics::Analysis of PDEs, fast diffusion equation, rates of convergence, stability, Harnack Principle, 26D10, 46E35, 35K55, 49J40, 35B40, 49K20, 49K30, 35J20, Mathematics - Analysis of PDEs, Hardy-Poincaré inequalities, spectral gap, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], self-similar Barenblatt solutions, asymptotic behavior, 2020 Mathematics Subject Classification.26D10, 46E35, 35K55, 49J40, 35B40, 49K20, 49K30, 35J20, Gagliardo-Nirenberg inequality, Analysis of PDEs (math.AP), entropy methods
Abstract

The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard Sobolev inequality (in dimension larger than three), or Onofri's inequality in dimension two. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. The regularity properties of the parabolic flow allow us to connect an improved entropy - entropy production inequality during an initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymptotic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms (a Fisher information which can be interpreted as a generalized Heisenberg uncertainty principle) the distance to the manifold of optimal functions. The method is constructive and, for the first time, quantitative estimates of the stability constant are obtained, including in the critical case of Sobolev's inequality. To build the estimates, we establish a quantitative global Harnack principle and perform a detailed analysis of large time asymptotics by entropy methods., Comment: This manuscript merges 2007.03674: Stability in Gagliardo-Nirenberg inequalities and 2007.03419: Stability in Gagliardo-Nirenberg inequalities. Supplementary material with several additions including: Chapter 1 (variational methods) and Chapter 6 (Sobolev's inequality)

Published
2021

8. Stability in Gagliardo-Nirenberg inequalities

Author

Bonforte, Matteo, Dolbeault, Jean, Nazaret, Bruno, Simonov, Nikita, Universidad Autonoma de Madrid (UAM), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Université Paris 1 Panthéon-Sorbonne (UP1), Fédération Parisienne de Modélisation Mathématique (FP2M), Centre National de la Recherche Scientifique (CNRS), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), This work has been partially supported by the Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR). M.B. and N.S. were partially supported by Projects MTM2017-85757-P (Spain) and by the E.U. H2020 MSCA programme, grant agreement 777822. N.S. was partially funded by the FPI-grant BES-2015-072962, associated to the project MTM2014-52240-P (Spain)., and ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017)

Subjects
Fast diffusion equation, Self-similar Barenblatt so-lutions, Entropy methods, Rates of convergence, Spectral gap, Hardy-Poincaré inequalities, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Intermediate asymptotics, 2020 Mathematics Subject Classification.26D10, 46E35, 35K55, 49J40, 35B40, 49K20, 49K30, 35J20, Stability, Harnack Principle, Gagliardo-Nirenberg inequality, Asymptotic behavior
Abstract

The purpose of this paper is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg inequalities. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. In both cases, the tail behaviour plays a key role. The regularity properties of the parabolic flow allow us to connect an improved entropy-entropy production inequality during the initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymp-totic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms the distance to the manifold of optimal functions.

Published
2020

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

Author

Matteo Muratori, Bruno Nazaret, Matteo Bonforte, Jean Dolbeault, Departamento de Matemáticas, Universidad Autonoma de Madrid (UAM), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Dipartimento di Matematica, 'Francesco Brioschi', Politecnico di Milano [Milan] (POLIMI), Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Université Paris 1 Panthéon-Sorbonne (UP1), ANR-13-BS01-0004,KIBORD,Modèles cinétiques en biologie et domaines connexes(2013), and ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012)

Subjects
Diffusion equation, Parabolic regularity, Spectral gap, Uniform convergence, Optimal functions, Linearization, 01 natural sciences, Mathematics - Analysis of PDEs, Weights, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Symmetry breaking, 0101 mathematics, Free energy, Best constants, Mathematics, Harnack's inequality, Harnack inequality, Numerical Analysis, Entropy methods, 010102 general mathematics, 35K55, 35B40, 49K30, 26D10, 35B06, 46E35, 49K20, 35J20, Asymptotic behavior, Caffarelli-Kohn- Nirenberg inequalities, Fast diffusion equation, Hardy-Poincaré inequalities, Intermediate asymptotics, Rate of convergence, Self-similar solutions, Modeling and Simulation, Caffarelli-Kohn-Nirenberg inequalities, 010101 applied mathematics, Symmetric function, Exponent, Analysis of PDEs (math.AP)
Abstract

International audience; 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 purpose of Part~II is 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 will be emphasized.

Published
2017

10. Fast diffusion equations: matching large time asymptotics by relative entropy methods

Author

Jean Dolbeault, Giuseppe Toscani, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Department of Mathematics, University of Pavia, and ANR-08-BLAN-0333,CBDif-fr,Collective behaviour & diffusion: mathematical models and simulations(2008)

Subjects
Change of variables, Kullback–Leibler divergence, Diffusion equation, intermediate asymptotics, Second moment of area, 01 natural sciences, Mathematics - Analysis of PDEs, FOS: Mathematics, Range (statistics), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Diffusion (business), Scaling, Mathematics, Numerical Analysis, asymptotic expansion, 010102 general mathematics, Mathematical analysis, optimal constants, second moment, 010101 applied mathematics, Fast diffusion equation, 35B40, 35K55, 39B62, sharp rates, Barenblatt solutions, large time behaviour, Modeling and Simulation, Hardy-Poincaré inequalities, porous media equation, Asymptotic expansion, Analysis of PDEs (math.AP)
Abstract

International audience; A non self-similar change of coordinates provides improved matching asymptotics of the solutions of the fast diffusion equation for large times, compared to already known results, in the range for which Barenblatt solutions have a finite second moment. The method is based on relative entropy estimates and a time-dependent change of variables which is determined by second moments, and not by the scaling corresponding to the self-similar Barenblatt solutions, as it is usually done.

Published
2011

11. Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities

Author

Matteo Bonforte, Juan Luis Vázquez, Jean Dolbeault, Gabriele Grillo, Departamento de Matemáticas, Universidad Autonoma de Madrid (UAM), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Dipartimento di Matematica 'Giuseppe Peano' [Torino], Università degli studi di Torino (UNITO), and ANR-08-BLAN-0333,CBDif-fr,Collective behaviour & diffusion: mathematical models and simulations(2008)

Subjects
Diffusion equation, intermediate asymptotics, 01 natural sciences, Mathematics - Analysis of PDEs, Convergence (routing), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Nonlinear diffusion, 0101 mathematics, Mathematics, Multidisciplinary, asymptotic expansion, Series (mathematics), 010102 general mathematics, Mathematical analysis, State (functional analysis), optimal constants, 010101 applied mathematics, Fast diffusion equation, Nonlinear system, sharp rates, Barenblatt solutions, large time behaviour, Hardy-Poincaré inequalities, Physical Sciences, porous media equation, Asymptotic expansion, Analysis of PDEs (math.AP)
Abstract

International audience; The goal of this note is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy-Poincaré inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities and rates for nonlinear diffusion equations.

Published
2010

12. Asymptotics of the fast diffusion equation via entropy estimates

Author

Juan Luis Vázquez, Matteo Bonforte, Adrien Blanchet, Gabriele Grillo, Jean Dolbeault, CRM, Institució Catalana de Recerca i Estudis Avançats (ICREA), Dipartimento di Matematica 'Giuseppe Peano' [Torino], Università degli studi di Torino (UNITO), Departamento de Matemáticas, Universidad Autonoma de Madrid (UAM), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
Diffusion equation, self-similar solutions, 01 natural sciences, Mathematics (miscellaneous), Mathematics - Analysis of PDEs, free energy methods, FOS: Mathematics, Entropy (information theory), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], asymptotic behavior, 0101 mathematics, B- ECONOMIE ET FINANCE, Mathematical physics, Physics, Euclidean space, Mechanical Engineering, 010102 general mathematics, 35B40, 010101 applied mathematics, Fast diffusion equation, 35K55, 39B62, Extinction time, Hardy-Poincaré inequalities, Analysis, Analysis of PDEs (math.AP)
Abstract

We consider non-negative solutions of the fast diffusion equation in the Euclidean space R^d, and study the asymptotic behavior of a natural class of solutions foir large times, where "large" means that t is approaching infinity in case the parameter m appearing in the equation is close to one, or that t approaches the finite extinction time in case the diffusion is "very fast". For a class of initial data, we prove that the solution converges with a polynomial rate to a self-similar solution. Such results are new in the "very fast" case, whereas when m is close to one we improve on known results. A fundamental role in this studied is played by suitable new functional inequalities which are related to the spectral properties of the linearized generator of the evolution considered, after suitable rescaling.

Published
2009

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