Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations

Matteo Bonforte,Juan Luis Vazquez

doi:10.1155/2007/21425

Abstract

We investigate local and global properties of positive solutions to the fast diffusion equation in the good exponent range , corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space , we prove sharp local positivity estimates (weak Harnack inequalities) and elliptic Harnack inequalities; also a slight improvement of the intrinsic Harnack inequality is given. We use them to derive sharp global positivity estimates and a global Harnack principle. Consequences of these latter estimates in terms of fine asymptotics are shown. For the mixed initial and boundary value problem posed in a bounded domain of with homogeneous Dirichlet condition, we prove weak, intrinsic, and elliptic Harnack inequalities for intermediate times. We also prove elliptic Harnack inequalities near the extinction time, as a consequence of the study of the fine asymptotic behavior near the finite extinction time.

Highlights

In this paper, we are interested in the questions of boundedness, positivity, and regularity of the solutions of fast diffusion equations
We investigate the connection between the global Harnack principle and the fine asymptotic behavior, first introduced by one of the authors in [1], in terms of uniform convergence in relative error, shortly CRE
The methods we present here may open new directions to solve the problem of Harnack inequalities for more general nonlinear diffusion equations for a larger range of exponents m

Summary

Introduction

We are interested in the questions of boundedness, positivity, and regularity of the solutions of fast diffusion equations. The combination of direct and reverse smoothing effects implies a Harnack inequality of elliptic type, compare Theorems 6.1 and 6.5, namely, we compare the supremum and infimum of the solution at the same time Another issue that we address is the extension to the whole space (or domain) of local positivity properties. This leads to the global Harnack principle, GHP, that is, to accurate upper and lower bounds in terms of some special (sub/super) solutions. The methods we present here may open new directions to solve the problem of Harnack inequalities for more general nonlinear diffusion equations for a larger range of exponents m.

Positivity results for the fast diffusion equation

Global Harnack principle on the whole space and relative error estimates

Convergence in relative error on a domain

Harnack inequalities