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.

We consider nonlinear diffusive evolution equations posed on bounded space domains, governed by fractional Laplace-type operators, and involving porous medium type nonlinearities. We establish existence and uniqueness results in a suitable class of solutions using the theory of maximal monotone operators on dual spaces. Then we describe the long-time asymptotics in terms of separate-variables solutions of the friendly giant type. As a by-product, we obtain an existence and uniqueness result for semilinear elliptic non local equations with sub-linear nonlinearities. The Appendix contains a review of the theory of fractional Sobolev spaces and of the interpolation theory that are used in the rest of the paper.

We investigate quantitative properties of the nonnegative solutions u(t, x) ≥ 0 to the nonlinear fractional diffusion equation, ∂ t u + L(u m ) = 0, posed in a bounded domain, x ∈ Ω ⊂ R N with m > 1 for t > 0. As L we use one of the most common definitions of the fractional Laplacian (−∆) s , 0 < s < 1, in a bounded domain with zero Dirichlet boundary conditions. We consider a general class of very weak solutions of the equation, and obtain a priori estimates in the form of smoothing effects, absolute upper bounds, lower bounds, and Harnack inequalities. We also investigate the boundary behaviour and we obtain sharp estimates from above and below. The standard Laplacian case s = 1 or the linear case m = 1 are recovered as limits. The method is quite general, suitable to be applied to a number of similar problems.

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