Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

TL;DR

This paper investigates symmetry and symmetry breaking in weighted fast diffusion equations related to Caffarelli-Kohn-Nirenberg inequalities, establishing conditions for symmetry breaking and analyzing asymptotic behaviors without assuming symmetry.

Contribution

It introduces explicit symmetry breaking conditions via linear instability analysis and links these to entropy inequalities and asymptotic rates in weighted fast diffusion equations.

Findings

Symmetry breaking occurs when radial optimal functions are linearly unstable.

Asymptotic convergence rates are characterized by weighted Hardy-Poincaré inequalities.

Linearized spectral analysis around self-similar profiles is key to understanding symmetry properties.

Abstract

In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not ? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy - entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN). We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy - entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincar{\'e} inequality which is interpreted as a linearized entropy - entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods. Consequences for the (WFD) flow will be studied in Part II of this work.