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.

We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we obtain a sharp form of the smoothing effect on such manifolds. We also estimate the location of the free boundary. A global Harnack principle follows.We also present a change of variables that allows to transform radially symmetric solutions of the PME on model manifolds into radially symmetric solutions of a corresponding weighted PME on Euclidean space and back. This equivalence turns out to be an important tool of the theory.

Abstract. We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian −∆ H N − (N − 1) 2 /4 on the hyperbolic space H N , (N − 1) 2 /4 being, as it is well-known, the bottom of the L 2 -spectrum of −∆ H N . We find the optimal constant in a resulting Poincaré-Hardy inequality, which includes a further remainder term which makes it sharp also locally: the resulting operator is in fact critical in the sense of [17]. A related improved Hardy inequality on more general manifolds, under suitable curvature assumption and allowing for the curvature to be possibly unbounded below, is also shown. It involves an explicit, curvature dependent and typically unbounded potential, and is again optimal in a suitable sense. Furthermore, with a different approach, we prove Rellich-type inequalities associated with the shifted Laplacian, which are again sharp in suitable senses.

We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ut = ∆ (u m /m) = div (u m−1 ∇u) posed for x ∈ R d , t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≥ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study [7] since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution.The linearization of this flow is interpreted here as the heat flow of the Laplace-Beltrami operator of a suitable Riemannian Manifold (R d , g), with a metric g which is conformal to the standard R d metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo-Nirenberg inequalities associated to the generator. Such inequalities in turn allow to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker-Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.