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 that yields a nonlinear 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 and provides answers to some open problems.

Rate of convergence to separable solutions of the fast diffusion equation

“…Depending on the decay rate of the initial function u 0 , other types of asymptotic behaviour near extinction may occur, such as convergence to Barenblatt profiles (see [3], [4], [5], [8], [9], [10], for example) or convergence to selfsimilar solutions of the second type (see [12], [22]).…”

Slow growth of solutions of superfast diffusion equations with unbounded initial data

“…Beyond this, the literature has provided more detailed information on how the asymptotic behaviour near extinction depends on the initial spatial decay, again indicating an important role of self-similar solutions (see e.g. [2], [3], [4], [5], [11], [10], [14], [16], [20]).…”

Optimal rates of convergence to the singular Barenblatt profile for the fast diffusion equation

“…The exponent m * plays an important role in the results on asymptotic behaviour near extinction in [1,3,4,6,[10][11][12]. The book [15] contains a general description of the phenomenon of extinction, even for m 0.…”