Rates of convergence to non-degenerate asymptotic profiles for fast diffusion via energy methods

Abstract

This paper is concerned with a quantitative analysis of asymptotic behaviors of (possibly sign-changing) solutions to the Cauchy-Dirichlet problem for the fast diffusion equation posed on bounded domains with Sobolev subcritical exponents. More precisely, rates of convergence to non-degenerate asymptotic profiles will be revealed via an energy method. The sharp rate of convergence to positive ones was recently discussed by Bonforte and Figalli [10] based on an entropy method. An alternative proof for their result will also be provided.

Regularity of solutions to the Dirichlet problem for fast diffusion equations

“…(iv) In the subcritical regime, Bonforte-Grillo-Vázquez [8] proved the uniform convergence of the relative error, and Bonforte-Figalli [7] proved the sharp exponential decay of the relative error on generic domains; see Akagi [1] for another proof. In our recent papers [37,38], we proved the sharp exponential decay of the relative error in the C 2 topology on generic domains, and polynomial decay of the relative error in the C 2 topology on all smooth domains.…”

“…In the pioneering work [6], Berryman and Holland studied the extinction behavior of solutions to the Cauchy-Dirichlet problem (1)- (3). Their work was motivated by the Wisconsin octupole experiments of Drake-Greenwood-Navratil-Post [31] on anomalous diffusion of hydrogen plasma across a purely poloidal octupole magnetic field, where the density of the plasma satisfies the fast diffusion equation (1) with p = 2. The experiments in [31] have shown that after a few milliseconds the solution u evolves into a fixed shape which then decays in time.…”

“…In the setting of linear uniformly parabolic equations, boundary estimates of type ( 6) have been studied in, e.g., Krylov [41], Fabes-Garofalo-Salsa [32] and Fabes-Safonov [33]. For boundary estimates of solutions to certain degenerate or singular nonlinear parabolic equations related to (1), we refer to the recent papers Kuusi-Mingione-Nyström [42], Avelin-Gianazza-Salsa [5] and the references therein. In particular, Avelin-Gianazza-Salsa [5] studied Carleson estimates and boundary Harnack inequality for local solutions of ( 1) and ( 2) with 1 < p < n (n−2) + , and it is mentioned in its Remark 4.10 that Hölder estimates for the gradient of the solutions, as well as (6), were still unknown.…”

Asymptotics Near Extinction for Nonlinear Fast Diffusion on a Bounded Domain

“…Their derivation was subsequently simplified by Agaki [2], though the comparison of his bounds in terms of an energy rather than the relative error exploits Jin and Xiong's recent boundary regularity theory [19]. On arbitrary smooth domains, however, the kernel is not necessarily trivial, hence the limiting stationary solution is in general not isolated, and thus, the proposed method is not applicable.…”