Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

Abstract

We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation u t = ∆u m , posed in a smooth bounded domain Ω ⊂ R N , in the exponent range m s = (N −2) + /(N +2) < m < 1. It is known that bounded positive solutions extinguish in a finite time T > 0, and also that they approach a separate variable solution u(t, x) ∼ (T − t) 1/(1−m) S(x), as t → T − . It has been shown recently that v(x, t) = u(t, x) (T − t) −1/(1−m) tends to S(x) as t → T − , uniformly in the relative error norm. Starting from this result, we investigate the fine asymptotic behaviour and prove sharp rates of convergence for the relative error. The proof is based on an entropy method relying on a (improved) weighted Poincaré inequality, that we show to be true on generic bounded domains. Another essential aspect of the method is the new concept of "almost orthogonality", which can be thought as a nonlinear analogous of the classical orthogonality condition needed to obtain improved Poincaré inequalities and sharp convergence rates for linear flows. Consider the Cauchy-Dirichlet problem for the Fast diffusion Equation (FDE)where m ∈ (0, 1), u 0 ≥ 0, and Ω ⊂ R N is a smooth bounded domain of class C 2,α . The main goal of this paper is to study the fine asymptotic behaviour of nonnegative solutions to this problem.This problem has been addressed for the first time in the '80s by Berryman and Holland in their pioneering work [6]. However, the results of [6] were not conclusive and many question were left open, due to the several difficulties hidden in this apparently simple problem. After that work, only a few relevant improvements appeared, and many basic questions are still open in many relevant aspects. We will give an account of the previous results concerning the problem under consideration, starting by a L ∞ (Ω) ≤ c ′ 0 e −t for all t ≫ 1

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.…”

“…Let p ≥ 1. Suppose (7) and (8) hold. Suppose w is Lipschitz continuous in B + 1 × (−1, 0], satisfies (10), and is a solution of (9) in the sense of distribution.…”

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

“…of convergence for non-degenerate (see below) positive asymptotic profiles was first discussed in [10] by developing the so-called nonlinear entropy method. We also refer the reader to recent works [19,20].…”

“…We also refer the reader to recent works [19,20]. Throughout this paper, as in [10], we assume that φ is non-degenerate, i.e., the linearized problem L φ (u) := −∆u − λ q (q − 1)|φ| q−2 u = 0 admits no non-trivial solution (or equivalently, L φ does not have zero eigenvalue), and hence, L φ is invertible. Then φ is also isolated in H 1 0 (Ω) from the other solutions to (1.7), (1.8).…”

Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type

“…This is already sufficient to accurately describe the asymptotics and study asymptotic profiles.If, however, ∂J is a non-linear and potentially multi-valued operator, the situation becomes more challenging since there is typically no basis of eigenfunctions available. Still there is a vast amount of literature dealing with the asymptotic behavior of certain partial differential equations like p-Laplacian equations [25,3,4,31,41], porous medium equations [39,35], fast diffusion equations [5,8,7], and other PDEs [20,6], the list far from being exhaustive. A common property of solutions to all this equations appears to be that asymptotically they behave like eigenfunctions of the associated operator.…”