Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods

Abstract

This paper is the second part of the study. In Part I, self-similar solutions of a weighted fast diffusion equation (WFD) were related to optimal functions in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities (CKN) applied to radially symmetric functions. For these inequalities, the linear instability (symmetry breaking) of the optimal radial solutions relies on the spectral properties of the linearized evolution operator. Symmetry breaking in (CKN) was also related to large-time asymptotics of (WFD), at formal level. A first purpose of Part II is to give a rigorous justification of this point, that is, to determine the asymptotic rates of convergence of the solutions to (WFD) in the symmetry range of (CKN) as well as in the symmetry breaking range, and even in regimes beyond the supercritical exponent in (CKN). Global rates of convergence with respect to a free energy (or entropy) functional are also investigated, as well as uniform convergence to self-similar solutions in the strong sense of the relative error. Differences with large-time asymptotics of fast diffusion equations without weights are emphasized

Weighted interpolation inequalities: a perturbation approach

“…Key ideas are provided by the weighted fast diffusion equation whose large time asymptotics are governed by the linearized problem. These asymptotics are studied in [9]. Recent progresses on the issue of symmetry breaking, which partially rely on the present paper, have been achieved in [25].…”

Caccioppoli-type estimates and Hardy-type inequalities derived from weighted p-harmonic problems

“…The best constants in the case p = 2 have been obtained in [13], where also the whole spectrum of the associated operator has been explicitly calculated, by means of linear spectral methods, different from the one we use here. In the case of Caffarelli-Kohn-Nirenberg weights we refer to [14,15,29] for analogous results.…”

“…Hardy inequalities may be used to construct other inequalities. Classical Hardy [45] Proven in [55] Hardy-Poincaré [13,25] Proven with explicitly computed spectrum of the associated linear operator [14,15] In…”

“…The symmetry and the phenomenon of symmetry breaking together with their consequences for optimal constants is studied in [27][28][29][30][31]. The applications of these type of inequalities to investigations on sharp asymptotics for related fast diffusion equations can be found in [14,15], whereas to quantitative regularity results for the related evolution equation in [16].…”

Quantitative a priori estimates for fast diffusion equations with Caffarelli–Kohn–Nirenberg weights. Harnack inequalities and Hölder continuity

“…see Subsection 1.3 for more details; these inequalities are deeply connected with the above WFDE, in its evolutionary or stationary version, see for instance [8,9,21,35,36,37,38]; some further connection will be discussed and explored in this paper.…”