Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation with a Critical Exponent

Abstract

We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a critical exponent. After a suitable rescaling which yields a non--linear Fokker--Planck equation, we find a continuum of algebraic rates of convergence to a self--similar profile. These rates depend explicitly on the spatial decay rates of initial data. This improves a previous result on slow convergence for the critical fast diffusion equation ({\sc Bonforte et al}. in Arch Rat Mech Anal 196:631--680, 2010) and provides answers to some open problems.

References in corpus (4)

• Asymptotics of the fast diffusion equation via entropy estimates
• Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities
• Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold
• Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation