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We consider a diffused interface version of the volume-preserving mean curvature flow in the Euclidean space, and prove, in every dimension and under natural assumptions on the initial datum, exponential convergence towards single "diffused balls"., Comment: 51 pages

We study nonnegative solutions to the Cauchy problem for the Fractional Fast Diffusion Equation on a suitable class of connected, noncompact Riemannian manifolds. This parabolic equation is both singular and nonlocal: the diffusion is driven by the (spectral) fractional Laplacian on the manifold, while the nonlinearity is a concave power makes the diffusion singular, so that solutions lose mass and may extinguish in finite time. Existence of mild solutions follows by nowadays standard nonlinear semigroups techniques, and we use these solutions as the building blocks for a more general class of so-called weak dual solutions, which allow for data both in the usual $L^1$ space and in a larger weighted space, determined in terms of the fractional Green function. We focus in particular on a priori smoothing estimates (also in weighted $L^p$ spaces) for a quite large class of weak dual solutions. We also show pointwise lower bounds for solutions, showing in particular that solutions have infinite speed of propagation. Finally, we start the study of how solutions extinguish in finite time, providing suitable sharp extinction rates.

Our focus is on the fast diffusion equation $\partial_t u=\Delta_p u$ with $p<2$ in the whole Euclidean space of dimension $N\geq 2$. The properties of the solutions to the $p$-Laplace Cauchy problem change in several special values of the parameter $p$. In the range of $p$ when mass is conserved, non-negative and integrable solutions behave like the Barenblatt (or fundamental) solutions for large times. By making use of the entropy method, we establish the polynomial rates of the convergence in the uniform relative error for a natural class of initial data. The convergence was established in the literature for $p$ close to $2$, but no rates were available. In particular, we allow for the values of $p$, for which the entropy is not displacement convex, as we do not apply the optimal transportation tools. We approach the issue of long-term asymptotics of the gradients of solutions. In fact, in the case of the radial initial datum, we provide also polynomial rates of the uniform convergence in the relative error of radial derivatives of solutions for $\frac{2N}{N+2}

We study regularity properties of solutions to nonlinear and nonlocal evolution problems driven by the so-called \emph{$0$-order fractional $p-$Laplacian} type operators: $$ \partial_t u(x,t)=\mathcal{J}_p u(x,t):=\int_{\mathbb{R}^n} J(x-y)|u(y,t)-u(x,t)|^{p-2}(u(y,t)-u(x,t))\,dy\,, $$ where $n\ge 1$, $p>1$, $J\colon\mathbb{R}^n\to\mathbb{R}$ is a bounded nonnegative function with compact support, $J(0)>0$ and normalized such that $\|J\|_{\mathrm{L}^1(\mathbb{R}^n)}=1$, but not necessarily smooth. We deal with Cauchy problems on the whole space, and with Dirichlet and Neumann problems on bounded domains. Beside complementing the existing results about existence and uniqueness theory, we focus on sharp regularity results in the whole range $p\in (1,\infty)$. When $p>2$, we find an unexpected $\mathrm{L}^q-\mathrm{L}^\infty$ regularization: the surprise comes from the fact that this result is false in the linear case $p=2$. We show next that bounded solutions automatically gain higher time regularity, more precisely that $u(x,\cdot)\in C^p_t$. We finally show that solutions preserve the regularity of the initial datum up to certain order, that we conjecture to be optimal ($p$-derivatives in space). When $p>1$ is integer we can reach $C^\infty$ regularity (gained in time, preserved in space) and even analyticity in time. The regularity estimates that we obtain are quantitative and constructive (all computable constants), and have a local character, allowing us to show further properties of the solutions: for instance, initial singularities do not move with time. We also study the asymptotic behavior for large times of solutions to Dirichlet and Neumann problems. Our results are new also in the linear case and are sharp when $p$ is integer. We expect them to be optimal for all $p>1$, supporting this claim with some numerical simulations., Comment: 53 pages

We consider a class of porous medium type of equations with Caputo time derivative. The prototype problem reads as $\Dc u=-\A u^m$ and is posed on a bounded Euclidean domain $\Omega\subset\mathbb{R}^N$ with zero Dirichlet boundary conditions. The operator $\A$ falls within a wide class of either local or nonlocal operators, and the nonlinearity is allowed to be of degenerate or singular type, namely, $01$. This equation is the most general form of a variety of models used to describe anomalous diffusion processes with memory effects, and finds application in various fields, including visco-elastic materials, signal processing, biological systems and geophysical science. We show existence of unique solution and new $L^p-L^\infty$ smoothing effects. The comparison principle, which we provide in the most general setting, serves as a crucial tool in the proof and provides a novel monotonicity formula. Consequently, we establish that the regularizing effects from the diffusion are stronger than the memory effects introduced by the fractional time derivative. Moreover, the solution attains the boundary conditions pointwise. Finally, we prove that the solution does not vanish in finite time if $00$. Our findings indicate that memory effects weaken the spatial diffusion and mitigate the difference between slow and fast diffusion.

The Fast Diffusion Equation (FDE) $u_t= \Delta u^m$, with $m\in (0,1)$, is an important model for singular nonlinear (density dependent) diffusive phenomena. Here, we focus on the Cauchy-Dirichlet problem posed on smooth bounded Euclidean domains. In addition to its physical relevance, there are many aspects that make this equation particularly interesting from the pure mathematical perspective. For instance: mass is lost and solutions may extinguish in finite time, merely integrable data can produce unbounded solutions, classical forms of Harnack inequalities (and other regularity estimates) fail to be true, etc. In this paper, we first provide a survey (enriched with an extensive bibliography) focussing on the more recent results about existence, uniqueness, boundedness and positivity (i.e., Harnack inequalities, both local and global), and higher regularity estimates (also up to the boundary and possibly up to the extinction time). We then prove new global (in space and time) Harnack estimates in the subcritical regime. In the last section, we devote a special attention to the asymptotic behaviour, from the first pioneering results to the latest sharp results, and we present some new asymptotic results in the subcritical case., Comment: 61 pages, 1 figure

We establish boundedness estimates for solutions of generalized porous medium equations of the form $$ \partial_t u+(-\mathfrak{L})[u^m]=0\quad\quad\text{in $\mathbb{R}^N\times(0,T)$}, $$ where $m\geq1$ and $-\mathfrak{L}$ is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, L\'evy operators. Our quantitative estimates take the form of precise $L^1$--$L^\infty$-smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of $-\mathfrak{L}$ and $I-\mathfrak{L}$. In the linear case $m=1$, it is well-known that the $L^1$--$L^\infty$-smoothing effect, or ultracontractivity, is equivalent to Nash inequalities. This is also equivalent to heat kernel estimates, which imply the Green function estimates that represent a key ingredient in our techniques. We establish a similar scenario in the nonlinear setting $m>1$. First, we can show that operators for which ultracontractivity holds, also provide $L^1$--$L^\infty$-smoothing effects in the nonlinear case. The converse implication is not true in general. A counterexample is given by $0$-order L\'evy operators like $-\mathfrak{L}=I-J\ast$. They do not regularize when $m=1$, but we show that surprisingly enough they do so when $m>1$, due to the convex nonlinearity. This reveals a striking property of nonlinear equations: the nonlinearity allows for better regularizing properties, almost independently of the linear operator. Finally, we show that smoothing effects, both linear and nonlinear, imply families of inequalities of Gagliardo-Nirenberg-Sobolev type, and we explore equivalences both in the linear and nonlinear settings through the application of the Moser iteration., Comment: 73 pages, 5 figures. v2: Updated according to the referee's suggestions. To appear in "Journal of Functional Analysis"

We study the homogeneous Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form $\partial_t u =-\mathcal{L} u^m$ posed on a bounded Euclidean domain $\Omega\subset\mathbb{R}^N$ with smooth boundary and $N\ge 1$. The linear diffusion operator $\mathcal{L}$ is a sub-Markovian operator, allowed to be of nonlocal type, while the nonlinearity is of singular type, namely $u^m=|u|^{m-1}u$ with $0

We provide a scheme of a recent stability result for a family of Gagliardo-Nirenberg-Sobolev (GNS) inequalities, which is equivalent to an improved entropy - entropy production inequality associated with an appropriate fast diffusion equation (FDE) written in self-similar variables. This result can be rephrased as an improved decay rate of the entropy of the solution of (FDE) for well prepared initial data. There is a family of Caffarelli-Kohn-Nirenberg (CKN) inequalities which has a very similar structure. When the exponents are in a range for which the optimal functions for (CKN) are radially symmetric, we investigate how the methods for (GNS) can be extended to (CKN). In particular, we prove that the solutions of the evolution equation associated to (CKN) also satisfy an improved decay rate of the entropy, after an explicit delay. However, the improved rate is obtained without assuming that initial data are well prepared, which is a major difference with the (GNS) case.

We study nonnegative solutions to the Fractional Porous Medium Equation on a suitable class of connected, noncompact Riemannian manifolds. We provide existence and smoothing estimates for solutions, in an appropriate weak (dual) sense, for data belonging either to the usual $L^1$ space or to a considerably larger weighted space determined in terms of the fractional Green function. The class of manifolds for which the results hold include both the Euclidean and the hyperbolic spaces and even in the Euclidean situation involve a class of data which is larger than previously known one., Comment: To appear in Math. Annalen

We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast $p$-Laplacian evolution equation $u_t=\Delta_p u$ on the whole Euclidean space, in the so-called "good fast diffusion range" $\tfrac{2N}{N+1}

This document comes as supplementary material of the paper Stability in Gagliardo-Nirenberg inequalities by the same authors. It is intended to state a number of classical or elementary statements concerning constants and inequalities for which we are not aware of existing published material or expressions detailed enough for our purpose. We claim no originality on the theoretical results and rely on standard methods in most cases, except that we keep track of the constants and provide constructive estimates.

The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard Sobolev inequality (in dimension larger than three), or Onofri's inequality in dimension two. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. The regularity properties of the parabolic flow allow us to connect an improved entropy - entropy production inequality during an initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymptotic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms (a Fisher information which can be interpreted as a generalized Heisenberg uncertainty principle) the distance to the manifold of optimal functions. The method is constructive and, for the first time, quantitative estimates of the stability constant are obtained, including in the critical case of Sobolev's inequality. To build the estimates, we establish a quantitative global Harnack principle and perform a detailed analysis of large time asymptotics by entropy methods., Comment: This manuscript merges 2007.03674: Stability in Gagliardo-Nirenberg inequalities and 2007.03419: Stability in Gagliardo-Nirenberg inequalities. Supplementary material with several additions including: Chapter 1 (variational methods) and Chapter 6 (Sobolev's inequality)

We consider the nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual $L^p$ spaces or to larger (weighted) spaces determined either in terms of a ground state of $\Delta_{\mathbb{H}^n}$, or of the (fractional) Green's function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $L^1-L^\infty$ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples.

We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the Fast Diffusion Equation with weights (WFDE) $u_t=|x|^\gamma\mathrm{div}\left(|x|^{-\beta}\nabla u^m\right)$ posed on $(0,+\infty)\times\mathbb{R}^d$, with $d\ge 3$, in the so-called good fast diffusion range $m_c

We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation $u_t=\Delta u^m$, posed in a smooth bounded domain $\Omega\subset \mathbb{R}^N$, in the exponent range $m_s=(N-2)_+/(N+2)0$, and also that they approach a separate variable solution $u(t,x)\sim (T-t)^{1/(1-m)}S(x)$, as $t\to T^-$. It has been shown recently that $v(x,t)=u(t,x)\,(T-t)^{-1/(1-m)}$ tends to $S(x)$ as $t\to T^-$, 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 Poincar\'e 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 Poincar\'e inequalities and sharp convergence rates for linear flows., Comment: 36 pages

We study a priori estimates for a class of non-negative local weak solution to the weighted fast diffusion equation $u_t = |x|^{\gamma} \nabla\cdot (|x|^{-\beta} \nabla u^m)$, with $0 < m <1$ posed on cylinders of $(0,T)\times{\mathbb R}^N$. The weights $|x|^{\gamma}$ and $|x|^{-\beta}$, with $\gamma < N$ and $\gamma -2 < \beta \leq \gamma(N-2)/N$ can be both degenerate and singular and need not belong to the class $\mathcal{A}_2$, a typical assumption for this kind of problems. This range of parameters is optimal for the validity of a class of Caffarelli-Kohn-Nirenberg inequalities, which play the role of the standard Sobolev inequalities in this more complicated weighted setting. The weights that we consider are not translation invariant and this causes a number of extra difficulties and a variety of scenarios: for instance, the scaling properties of the equation change when considering the problem around the origin or far from it. We therefore prove quantitative - with computable constants - upper and lower estimates for local weak solutions, focussing our attention where a change of geometry appears. Such estimates fairly combine into forms of Harnack inequalities of forward, backward and elliptic type. As a consequence, we obtain H\"older continuity of the solutions, with a quantitative (even if non-optimal) exponent. Our results apply to a quite large variety of solutions and problems. The proof of the positivity estimates requires a new method and represents the main technical novelty of this paper. Our techniques are flexible and can be adapted to more general settings, for instance to a wider class of weights or to similar problems posed on Riemannian manifolds, possibly with unbounded curvature. In the linear case, $m=1$, we also prove quantitative estimates, recovering known results in some cases and extending such results to a wider class of weights., Comment: 53 pages, 2 figures. This is a much improved version of the older manuscript, thanks to the suggestions of the anonymous referee

We investigate quantitative properties of nonnegative solutions $u(x)\ge 0$ to the semilinear diffusion equation $\mathcal{L} u= f(u)$, posed in a bounded domain $\Omega\subset {\mathbb R}^N$ with appropriate homogeneous Dirichlet or outer boundary conditions. The operator $\mathcal{L}$ may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian $(-\Delta)^s$ ($0

This paper provides a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form $\partial_t u + {\mathcal L}u^m=0$, $m>1$, where the operator ${\mathcal L}$ belongs to a general class of linear operators, and the equation is posed in a bounded domain $\Omega\subset{\mathbb R}^N$. As possible operators we include the three most common definitions of the fractional Laplacian in a bounded domain with zero Dirichlet conditions, and also a number of other nonlocal versions. In particular, ${\mathcal L}$ can be a power of a uniformly elliptic operator with $C^1$ coefficients. Since the nonlinearity is given by $u^m$ with $m>1$, the equation is degenerate parabolic. The basic well-posedness theory for this class of equations has been recently developed in [14,15]. Here we address the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our results cover also the local case when ${\mathcal L}$ is a uniformly elliptic operator, and provide new estimates even in this setting. A surprising aspect discovered in this paper is the possible presence of non-matching powers for the long-time boundary behavior. More precisely, when ${\mathcal L}=(-\Delta)^s$ is a spectral power of the {Dirichlet} Laplacian inside a smooth domain, we can prove that: - when $2s> 1-1/m$, for large times all solutions behave as ${\rm dist}^{1/m}$ near the boundary; - when $2s\le 1-1/m$, different solutions may exhibit different boundary behavior. This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the semilinear elliptic equation ${\mathcal L}u^m=u$., Comment: 36 pages, 3 figures (6 images). This is a much improved version of the older manuscripts, thanks to the suggestions of the anonymous referees. To appear in Analysis & PDE (2017)

We construct a theory of existence, uniqueness and regularity of solutions for the fractional heat equation $\partial_t u +(-\Delta)^s u=0$, $0

This paper is the second part of the study. In Part~I, self-similar solutions of a weighted fast diffusion equation (WFD) were related to optimal functions in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities (CKN) applied to radially symmetric functions. For these inequalities, the linear instability (symmetry breaking) of the optimal radial solutions relies on the spectral properties of the linearized evolution operator. Symmetry breaking in (CKN) was also related to large-time asymptotics of (WFD), at formal level. A first purpose of Part~II is to give a rigorous justification of this point, that is, to determine the asymptotic rates of convergence of the solutions to (WFD) in the symmetry range of (CKN) as well as in the symmetry breaking range, and even in regimes beyond the supercritical exponent in (CKN). Global rates of convergence with respect to a free energy (or entropy) functional are also investigated, as well as uniform convergence to self-similar solutions in the strong sense of the relative error. Differences with large-time asymptotics of fast diffusion equations without weights will be emphasized.

In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not ? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy - entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN). We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy - entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincar{\'e} inequality which is interpreted as a linearized entropy - entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods. Consequences for the (WFD) flow will be studied in Part II of this work.

We study the positivity and regularity of solutions to the fractional porous medium equations $u_t+(-\Delta)^su^m=0$ in $(0,\infty)\times\Omega$, for $m>1$ and $s\in (0,1)$ and with Dirichlet boundary data $u=0$ in $(0,\infty)\times({\mathbb R}^N\setminus\Omega)$, and nonnegative initial condition $u(0,\cdot)=u_0\geq0$. Our first result is a quantitative lower bound for solutions which holds for all positive times $t>0$. As a consequence, we find a global Harnack principle stating that for any $t>0$ solutions are comparable to $d^{s/m}$, where $d$ is the distance to $\partial\Omega$. This is in sharp contrast with the local case $s=1$, in which the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior ($C^\infty$ in $x$ and $C^{1,\alpha}$ in $t$) and establish a sharp $C^{s/m}_x$ regularity estimate up to the boundary. Our methods are quite general, and can be applied to a wider class of nonlocal parabolic equations of the form $u_t-\mathcal L F(u)=0$ in $\Omega$, both in bounded or unbounded domains.

We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L}F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$, with appropriate homogeneous Dirichlet boundary conditions. As $\mathcal{L}$ we can use a quite general class of linear operators that includes the two most common versions of the fractional Laplacian $(-\Delta)^s$, $01$. In this paper we propose a suitable class of solutions of the equation, and cover the basic theory: we prove existence, uniqueness of such solutions, and we establish upper bounds of two forms (absolute bounds and smoothing effects), as well as weighted-$L^1$ estimates. The class of solutions is very well suited for that work. The standard Laplacian case $s=1$ is included and the linear case $m=1$ can be recovered in the limit. In a companion paper [12], we will complete the study with more advanced estimates, like the upper and lower boundary behaviour and Harnack inequalities, for which the results of this paper are needed., Comment: arXiv admin note: text overlap with arXiv:1311.6997

We show non-existence of solutions of the Cauchy problem in $\mathbb{R}^N$ for the nonlinear parabolic equation involving fractional diffusion $\partial_t u + (-\Delta)^s \phi(u)= 0,$ with $00$, or $\phi(u) = \log u$, and we take nonnegative $L^1$ initial data, there is no (nonnegative) solution of the problem in any dimension $N\ge 2$. We find the range of non-existence when $N=1$ in terms of $s$ and $n$. The range of exponents that we find for non-existence both for parabolic and elliptic equations are optimal. Non-existence is then proved for more general nonlinearities $\phi$, and it is also extended to the related elliptic problem of nonlinear nonlocal type: $u + (-\Delta)^s \phi(u) = f$ with the same type of nonlinearity $\phi$., Comment: 29 pages, 1 figure

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., Comment: Keywords: Fractional Laplace operators, Porous Medium diffusion, Existence and uniqueness theory, Asymptotic behaviour, Fractional Sobolev Spaces

We investigate quantitative properties of the nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + {\mathcal L} (u^m)=0$, posed in a bounded domain, $x\in\Omega\subset {\mathbb R}^N$ with $m>1$ for $t>0$. As ${\mathcal L}$ we use one of the most common definitions of the fractional Laplacian $(-\Delta)^s$, $0

We establish quantitative estimates for solutions $u(t,x)$ to the fractional nonlinear diffusion equation, $\partial_t u +(-\Delta)^s (u^m)=0$ in the whole range of exponents $m>0$, $0

The purpose of this paper is to prove local upper and lower bounds for weak solutions of semilinear elliptic equations of the form $-\Delta u= c u^p$, with $0

We consider the dynamics of the Total Variation Flow (TVF) $u_t=\div(Du/|Du|)$ and of the Sign Fast Diffusion Equation (SFDE) $u_t=\Delta\sign(u)$ in one spatial dimension. We find the explicit dynamic and sharp asymptotic behaviour for the TVF, and we deduce the one for the SFDE by an explicit correspondence between the two equations.

We study the Emden-Fowler equation $-\Delta u=|u|^{p-1}u$ on the hyperbolic space ${\mathbb H}^n$. We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is $p=(n+2)/(n-2)$ as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers \cite{mancini, bhakta} consider finite energy solutions, we shall deal here with infinite energy solutions and we determine the exact asymptotic behavior of wide classes of finite and infinite energy solutions.

We consider the Fast Diffusion Equation $u_t=\Delta u^m$ posed in a bounded smooth domain $\Omega\subset \RR^d$ with homogeneous Dirichlet conditions; the exponent range is $m_s=(d-2)_+/(d+2)

The goal of this note 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-Poincar\'e inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities and rates for nonlinear diffusion equations.

We investigate qualitative properties of local solutions $u(t,x)\ge 0$ to the fast diffusion equation, $\partial_t u =\Delta (u^m)/m$ with $m<1$, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form $[0,T]\times\RR^d$. They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for low $m$ in the so-called very fast diffusion range, precisely for all $m\le m_c=(d-2)/d.$ The boundedness statements are true even for $m\le 0$, while the positivity ones cannot be true in that range., Comment: 36 pages, 1 figure

We consider the asymptotic behaviour of positive solutions $u(t,x)$ of the fast diffusion equation $u_t=\Delta (u^{m}/m)={\rm div} (u^{m-1}\nabla u)$ posed for $x\in\RR^d$, $t>0$, with a precise value for the exponent $m=(d-4)/(d-2)$. The space dimension is $d\ge 3$ so that $m<1$, and even $m=-1$ for $d=3$. This case had been left open in the general study \cite{BBDGV} 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 $(\RR^d,{\bf g})$, with a metric ${\bf g}$ which is conformal to the standard $\RR^d$ 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 $m$., Comment: 37 pages

We consider non-negative solutions of the fast diffusion equation $u_t=\Delta u^m$ with $m \in (0,1)$, in the Euclidean space R^d, d?3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to $t\to\infty$ for $m\ge m_c=(d-2)/d$, 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 $m\ge m_c$, or close enough to the extinction time if m < mc. Such results are new in the range $m\le m_c$ where previous approaches fail. In the range mc < m < 1 we improve on known results.