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

Abstract

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.

References in corpus (3)

• Asymptotics of the fast diffusion equation via entropy estimates
• Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold
• On the extinction profile of solutions to fast-diffusion

Cited by in corpus (28)

• One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: Remarks on duality and flows
• Radial Fast Diffusion on the Hyperbolic Space
• Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation
• Renyi entropy and improved equilibration rates to self-similarity for nonlinear diffusion equations
• Stability of non-isolated asymptotic profiles for fast diffusion
• Best matching Barenblatt profiles are delayed
• Modified log-Sobolev inequalities, Beckner inequalities and moment estimates
• Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method
• Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results
• Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation with a Critical Exponent
• Regularisation effects of nonlinear semigroups
• Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type
• A fractional Yamabe flow and some applications
• Weighted interpolation inequalities: a perturbation approach
• Extinction behaviour for the fast diffusion equations with critical exponent and Dirichlet boundary conditions
• L2-Hypocoercivity and large time asymptotics of the linearized Vlasov-Poisson-Fokker-Planck system
• Improved energy methods for nonlocal diffusion problems
• Multi-D Fast Diffusion Equation via Diffusive Scaling of Generalized Carleman Kinetic Equation
• Improved interpolation inequalities, relative entropy and fast diffusion equations
• A Continuum of Extinction Rates for the Fast Diffusion Equation
• Hypocoercivity for linear kinetic equations conserving mass
• Stability in Gagliardo-Nirenberg inequalities - Supplementary material
• Fractional diffusion for Fokker-Planck equation with heavy tail equilibrium: an à la Koch spectral method in any dimension
• First order Hardy inequalities revisited
• Asymptotic large time behavior of singular solutions of the fast diffusion equation
• A higher-order gradient flow scheme for a singular one-dimensional diffusion equation
• Existence of solutions to degenerate parabolic problems with two weights via the Hardy inequality
• Fast diffusion equations: matching large time asymptotics by relative entropy methods