Weighted fast diffusion equations I: sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

DOI10.3934/krm.2017002zbMath1361.35075arXiv1602.08319OpenAlexW2281813348MaRDI QIDQ728171

Bruno Nazaret, Matteo Muratori, Jean Dolbeault, Matteo Bonforte

Publication date: 19 December 2016

Published in: Kinetic and Related Models (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1602.08319

zbMATH Keywords

interpolationsymmetryweightsfast diffusion equationsemilinear elliptic equationslinearizationspectrumspectral gapflowsfunctional inequalitiessymmetry breakingentropy methodsCaffarelli-Kohn-Nirenberg inequalitiesbest constantsHardy-Poincaré inequalityoptimal functions

Mathematics Subject Classification ID

Optimality conditions for problems involving partial differential equations (49K20) Nonlinear parabolic equations (35K55) Sobolev spaces and other spaces of ``smooth functions, embedding theorems, trace theorems (46E35) Inequalities involving derivatives and differential and integral operators (26D10) Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) (49K30)

Related Items (16)

The Cauchy problem for the fast \(p\)-Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour ⋮ A pocket guide to nonlinear differential equations in Musielak-Orlicz spaces ⋮ Weighted interpolation inequalities: a perturbation approach ⋮ Existence to nonlinear parabolic problems with unbounded weights ⋮ Weighted Gagliardo-Nirenberg interpolation inequalities ⋮ Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization ⋮ Instantaneous and finite time blow-up of solutions to a reaction-diffusion equation with Hardy-type singular potential ⋮ Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations ⋮ Pointwise estimates of solutions to the weighted porous medium equation and the fast diffusion one via weighted Riesz potentials ⋮ Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli-Kohn-Nirenberg weights ⋮ Quantitative a priori estimates for fast diffusion equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Hölder continuity ⋮ Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities ⋮ On the continuity of solutions of the equations of a porous medium and the fast diffusion with weighted and singular lower-order terms ⋮ Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results ⋮ Caccioppoli-type estimates and Hardy-type inequalities derived from weighted \(p\)-harmonic problems ⋮ A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation

Cites Work

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• Nonlinear diffusions: extremal properties of Barenblatt profiles, best matching and delays
• Radial symmetry and symmetry breaking for some interpolation inequalities
• Fast diffusion equations: matching large time asymptotics by relative entropy methods
• Global positivity estimates and Harnack inequalities for the fast diffusion equation
• Potential theory and optimal convergence rates in fast nonlinear diffusion
• Hardy-Poincaré inequalities and applications to nonlinear diffusions
• Rigidity results with applications to best constants and symmetry of Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities
• Special fast diffusion with slow asymptotics: Entropy method and flow on a Riemannian manifold
• A logarithmic Hardy inequality
• Asymptotics of the fast diffusion equation via entropy estimates
• Weighted Poincaré-type inequalities for Cauchy and other convex measures
• Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions.
• Fast diffusion to self-similarity: Complete spectrum, long-time asymptotics, and numerology
• Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type.
• Entropies and equilibria of many-particle systems: an essay on recent research
• Improved interpolation inequalities, relative entropy and fast diffusion equations
• Kinetic approach to long time behavior of linearized fast diffusion equations
• Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces
• ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS
• Stability Results for Logarithmic Sobolev and Gagliardo–Nirenberg Inequalities
• Extremal functions for Caffarelli—Kohn—Nirenberg and logarithmic Hardy inequalities
• Poincaré inequalities for linearizations of very fast diffusion equations
• The Asymptotic Behavior of Gas in an n-Dimensional Porous Medium
• Asymptotic L^1-decay of solutions of the porous medium equation to self-similarity
• Fine Asymptotics for Fast Diffusion Equations
• Self‐similar Asymptotics for Linear and Nonlinear Diffusion Equations
• On Fast-Diffusion Equations with Infinite Equilibrium Entropy and Finite Equilibrium Mass
• Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities
• Best matching Barenblatt profiles are delayed
• Higher-order time asymptotics of fast diffusion in Euclidean space: A dynamical systems approach
• Phase transitions and symmetry breaking in singular diffusion
• Bounds for the Discrete Part of the Spectrum of a Semi-Bounded Schrödinger Operator.
• Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities

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