Abstract. We consider non-negative solutions of the fast diffusion equation ut = ∆u m with m ∈ (0, 1), in the Euclidean space R d , d ≥ 3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to t → ∞ for m ≥ mc = (d−2)/d, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≥ mc, or close enough to the extinction time if m < mc. Such results are new in the range m ≤ mc where previous approaches fail. In the range mc < m < 1 we improve on known results.

The goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy-Poincaré inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities, and rates for nonlinear diffusion equations. T he evolution equationwith m ≠ 1 is a simple example of a nonlinear diffusion equation which generalizes the heat equation and appears in a wide number of applications. Solutions differ from the linear case in many respects, notably concerning existence, regularity, and large-time behavior. We consider positive solutions uðτ; yÞ of this equation posed for τ ≥ 0 and y ∈ R d , d ≥ 1. The parameter m can be any real number. The equation makes sense even in the limit case m ¼ 0, where u m ∕m has to be replaced by log u, and is formally parabolic for all m ∈ R. Notice that [1] is degenerate at the level u ¼ 0 when m > 1 and singular when m < 1. We consider the initial-value problem with nonnegative datum uðτ ¼ 0; ·Þ ¼ u 0 ∈ L 1 loc ðdxÞ, where dx denotes Lebesgue's measure on R d . Further assumptions on u 0 are needed and will be specified later.The description of the asymptotic behavior of the solutions of [1] as τ → ∞ is a classical and very active subject. If m ¼ 1, the convergence of solutions of the heat equation with u 0 ∈ L 1 þ ðdxÞ to the Gaussian kernel (up to a mass factor) is a cornerstone of the theory. In the case of Eq. 1 with m > 1, known in the literature as the porous medium equation, the study of asymptotic behavior goes back to ref. On the other hand, when m < m c , a natural extension for the Barenblatt functions can be achieved by considering the same expression [2], but a different form for R, that is,The parameter T now denotes the extinction time, an important feature. The limit case m ¼ m c is covered by RðτÞ ¼ e τ , U D;T ðτ; yÞ ¼ e −dτ ðD þ e −2τ jyj 2 ∕dÞ −d∕2 . See refs. 4 and 5 for more detailed considerations. In this paper, we shall focus our attention on the case m < 1 which has been much less studied. In this regime, [1] is known as the fast diffusion equation. We do not even need to assume m > 0. We shall summarize and extend a series of recent results on the basin of attraction of the family of generalized Barenblatt solutions and establish the optimal rates of convergence of the solutions of [1] toward a unique attracting limit state in that family. Such basin of attraction is different according to m being above or below the value m à ≔ðd − 4Þ∕ðd − 2Þ, and for m ¼ m à the long-time behavior of the solutions has specific features. To state our results, it is more convenient to rescale the flow and rewrite

We systematically study weighted Poincaré type inequalities which are closely connected with Hardy type inequalities and establish the form of the optimal constants in some cases. Such inequalities are then used to relate entropy with entropy production and get intermediate asymptotics results for fast diffusion equations. To cite this article: A. Blanchet et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007). Résumé Inégalités de Hardy-Poincaré et applications. Nous étudions des inégalités de Poincaré qui sont étroitement reliées à des inégalités de type Hardy et établissons la forme des constantes optimales dans certains cas. De telles inégalités sont ensuite utilisées pour relier l'entropie avec la production d'entropie et obtenir des résultats d'asymptotiques intermédiaires pour les équations à diffusion rapide. Pour citer cet article : A. Blanchet et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

We consider the Fast Diffusion Equation ut = ∆u m posed in a bounded smooth domain Ω ⊂ R d with homogeneous Dirichlet conditions; the exponent range is ms = (d − 2)+/(d + 2) < m < 1. It is known that bounded positive solutions u(t, x) of such problem extinguish in a finite time T , and also that such solutions approach a separate variable solutionHere we are interested in describing the behaviour of the solutions near the extinction time. We first show that the convergence u(t, x) (T − t) −1/(1−m) to S(x) takes place uniformly in the relative error norm. Then, we study the question of rates of convergence of the rescaled flow. For m close to 1 we get such rates by means of entropy methods and weighted Poincaré inequalities. The analysis of the latter point makes an essential use of fine properties of the associated stationary elliptic problem −∆S m = cS in the limit m → 1, and such a study has an independent interest.

Let M be a compact Riemannian manifold without boundary. Consider the porous media equationu = (u m ), u(0) = u 0 ∈ L q , being the Laplace-Beltrami operator. Then, if q 2 ∨ (m − 1), the associated evolution is L q − L ∞ regularizing at any time t > 0 and the bound u(t) ∞ C(u 0 )/t holds for t < 1 for suitable explicit C(u 0 ), . For large t it is shown that, for general initial data, u(t) approaches its time-independent mean with quantitative bounds on the rate of convergence. Similar bounds are valid when the manifold is not compact, but u(t) approaches u ≡ 0 with different asymptotics. The case of manifolds with boundary and homogeneous Dirichlet, or Neumann, boundary conditions, is treated as well. The proof stems from a new connection between logarithmic Sobolev inequalities and the contractivity properties of the nonlinear evolutions considered, and is therefore applicable to a more abstract setting.