Behaviour near extinction for the Fast Diffusion Equation on bounded domains

Bonforte, Matteo; Grillo, Gabriele; Vázquez, Juan Luís

doi:10.1016/j.matpur.2011.03.002

Cited by 54 publications

(120 citation statements)

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“…However the results were not sharp and only applied to a strict subset of the range (m s , 1) not easy to quantify. For more details, see Sections 4 and 5 of [16] or also the last example at the end of Subsection 2.4. When dealing with strictly positive Dirichlet data, an entropy method similar to [16] has been developed in [8].…”

Section: (Rcdp)mentioning

confidence: 99%

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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%

“…follows for instance from [14, Theorem 5.9] (see also [33,16]) since V is a solution to the semilinear equation (EDP-V). Once the result for φ 1,1 is established, it suffices to note that |φ k,j | is a subsolution to the linear elliptic equation…”

Section: Short Notationmentioning

confidence: 99%

Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

Bonforte

Figalli

2020

Comm Pure Appl Math

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“…This peculiar feature comes from the fact that problem (2.1) admits solutions with extinction in finite time: u(·, t) → 0 uniformly as t → T , for some finite time T > 0 (cfr. with [8,10] for the fast Porous Medium framework and [9] for the fast p-Laplacian one) that could be employed as super-solutions to (1.1), obtaining that also solutions defined in tubes extinguish in finite time. Clearly, in this case the study of propagation of solutions has a very different nature.…”

Section: Comments and Open Problemsmentioning

confidence: 99%

Travelling wave behaviour arising in nonlinear diffusion problems posed in tubular domains

Audrito

Vázquez

2020

Journal of Differential Equations

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For a fixed bounded domain D ⊂ R N we investigate the asymptotic behaviour for large times of solutions to the p-Laplacian diffusion equation posed in a tubular domainwith p > 2, i.e., the slow diffusion case, and homogeneous Dirichlet boundary conditions on the tube boundary. Passing to suitable re-scaled variables, we show the existence of a travelling wave solution in logarithmic time that connects the level u = 0 and the unique nonnegative steady state associated to the re-scaled problem posed in a lower dimension, i.e. in D ⊂ R N .We then employ this special wave to show that a wide class of solutions converge to the universal stationary profile in the middle of the tube and at the same time they spread in both axial tube directions, miming the behaviour of the travelling wave (and its reflection) for large times.The first main feature of our analysis is that wave fronts are constructed through a (nonstandard) combination of diffusion and absorbing boundary conditions, which gives rise to a sort of Fisher-KPP long-time behaviour. The second one is that the nonlinear diffusion term plays a crucial role in our analysis. Actually, in the linear diffusion framework p = 2 solutions behave quite differently.

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“…Then, the extinction rate for bounded solutions is universal, of the form u(·, τ ) ∞ = O((T − τ ) 1/(1−m) ) when m > m s := (n − 2)/(n + 2), see [1], but the question is more complicated when m m s . Convergence rates for m near 1 have been recently obtained in [5].…”

Section: Commentsmentioning

confidence: 99%