Stability analysis of asymptotic profiles for sign-changing solutions to fast diffusion equations

Akagi, Goro; Kajikiya, Ryuji

doi:10.1007/s00229-012-0583-9

Cited by 14 publications

(31 citation statements)

References 48 publications

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“…Related Problems: signed solutions and subcritical range. In the case of signed solution the situation gets even more involved [1,2,3,4]. As for the subcritical range: the case m = m s corresponds to the celebrated Yamabe flow, many results have been obtained in different settings, but sharp asymptotic results are still missing for the Dirichlet problem.…”

Section: (Rcdp)mentioning

confidence: 99%

Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

Bonforte

Figalli

2020

Comm Pure Appl Math

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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

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Section: (Rcdp)mentioning

confidence: 99%

Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

Bonforte

Figalli

2020

Comm Pure Appl Math

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“…Then the function t → R(u(t)) is non-increasing, and hence, so is the function s → R(v(s)) (see, e.g., [6,30,36,1]).…”

Section: 1mentioning

confidence: 99%

“…On the other hand, from the continuity of t * : H 1 0 (Ω) → [0, ∞) (see [1,Proposition 4]) and the fact that t * (φ) = 1 by φ ∈ X , one deduces that v 0,µ → φ strongly in H 1 0 (Ω), whence v 0,µ belongs to B H 1 0 (Ω) (φ; r 0 ) for µ sufficiently close to 1. However, these facts contradict the local minimality of J at φ over X .…”

Section: Local Minimizers Of J Over Xmentioning

confidence: 99%

“…whenever v(·, 0) ∈ X and v(·, 0) − φ H 1 0 (Ω) < δ 0 . In [1], some stability criteria are also established for isolated profiles (see Proposition 1.2 below). Here least energy solutions to (1.8), (1.9) mean nontrivial solutions achieving the least energy, that is, the infimum over S of the energy functional J :…”

Section: Introductionmentioning

confidence: 99%

“…Proposition 1.2 (Stability criteria for isolated asymptotic profiles [1]). The following (i) and (ii) hold true:…”

Section: Introductionmentioning

confidence: 99%

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Stability of Non-Isolated Asymptotic Profiles for Fast Diffusion

Akagi

2016

Commun. Math. Phys.

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The stability of asymptotic profiles of solutions to the Cauchy-Dirichlet problem for fast diffusion equation (FDE, for short) is discussed. The main result of the present paper is the stability of any asymptotic profiles of least energy. It is noteworthy that this result can cover non-isolated profiles, e.g., those for thin annular domain cases. The method of proof is based on the Łojasiewicz-Simon inequality, which is usually used to prove the convergence of solutions to prescribed limits, as well as a uniform extinction estimate for solutions to FDE. Besides, local minimizers of an energy functional associated with this issue are characterized. Furthermore, the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the Łojasiewicz-Simon inequality in a different way.

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Stability and Instability of Group Invariant Asymptotic Profiles for Fast Diffusion Equations

Akagi¹

2013

Geometric Properties for Parabolic and Elliptic PDE's

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Stability analysis of asymptotic profiles for sign-changing solutions to fast diffusion equations

Cited by 14 publications

References 48 publications

Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

Stability of Non-Isolated Asymptotic Profiles for Fast Diffusion

Stability and Instability of Group Invariant Asymptotic Profiles for Fast Diffusion Equations

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