Matteo Bonforte
email: matteo.bonforte@uam.es

Jean Dolbeault
email: dolbeaul@ceremade.dauphine.fr

Matteo Muratori
email: matteo.muratori@unipv.it

Bruno Nazaret
email: bruno.nazaret@univ-paris1.fr

WEIGHTED FAST DIFFUSION EQUATIONS (PART II): SHARP ASYMPTOTIC RATES OF CONVERGENCE IN RELATIVE ERROR BY ENTROPY METHODS

Keywords: June 18, 2016. 2010 Mathematics Subject Classification. Primary: 35K55, 35B40, 49K30; Secondary: 26D10, 35B06, 46E35, 49K20, 35J20 Fast diffusion equation, self-similar solutions, asymptotic behavior, intermediate asymptotics, rate of convergence, entropy methods, free energy, Caffarelli-Kohn-Nirenberg inequalities, Hardy-Poincaré inequalities, weights, optimal functions, best constants, symmetry breaking, linearization, spectral gap, Harnack inequality, parabolic regularity

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 are emphasized.

Introduction. Let us consider the fast diffusion equation with weights

u t + |x| γ ∇ • |x| -β u ∇u m-1 = 0 , (t, x) ∈ R + × R d . (1)
Such an equation admits the self-similar solution

u (t, x) = ρ t ρ (d-γ) 1 + |( ρ t ) ρ x| 2+β-γ 1 m-1 , ∀ (t, x) ∈ R + × R d ,
where 1 ρ = (dγ) (mm c ) with m c := d-2-β d-γ . At least when 1m > 0 is not too big, this self-similar solution attracts all solutions to (1) as t → ∞, but we will also prove that, exactly as for the non-weighted equation corresponding to (β, γ) = (0, 0), there is a basin of attraction of u for any m ∈ (0, 1). However, there are many differences with respect to the non-weighted case, which will be summarized in Section 4.3. To study the convergence of u to u , it is simpler to use self-similar variables (see Section 2.3 for details) and consider the Fokker-Plancktype equation

v t + |x| γ ∇ • |x| -β v ∇ v m-1 -|x| 2+β-γ = 0 (2)
with initial condition v(t = 0, •) = v 0 . Self-similar solutions are transformed into Barenblatt-type stationary solutions given by

B(x) := C(M ) + |x| 2+β-γ 1 m-1 ,
where C(M ) is a positive constant uniquely determined by the weighted mass condition [START_REF] Blanchet | Hardy-Poincaré inequalities and applications to nonlinear diffusions[END_REF]. Altogether, what we aim at is establishing an exponential convergence of v to B as t → ∞, when the corresponding distance is measured in terms of the free energy (which is sometimes called generalized relative entropy in the literature)

R d B dx |x| γ = M := R d v 0 dx |x| γ , at least if m ∈ (m c ,
F[v] := 1 m -1 R d v m -B m -m B m-1 (v -B) dx |x| γ . (3)
By evolving such free energy along the flow and differentiating with respect to t, we formally obtain that

d dt F[v(t)] = - m 1 -m I[v(t)] , (4)
where I[v] denotes the relative Fisher information

I[v] := R d v ∇v m-1 -∇B m-1 2 dx |x| β .
This will be proved rigorously in Section 3.1. Note that, with some abuse of notation, when we write v(t) we mean the whole spatial profile of the function evaluated at time t.

Our goal is to relate F[v] and I[v], at least as t → ∞, and for that we need a detour by a family of Caffarelli-Kohn-Nirenberg inequalities which have been introduced in [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] and studied in Part I of this work, [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF]. Let us explain this a bit more in detail. For all q ≥ 1, consider the weighted norms

w L q,γ (R d ) := R d |w| q |x| -γ dx 1 q
with w L q (R d ) := w L q,0 (R d ) , and define L q,γ (R d ) as the space of all measurable functions w such that w L q,γ (R d ) is finite. Actually, at some points below, we shall also make use of the above definition for q ∈ (0, 1) (in which cases clearly w L q,γ (R d ) is no more a norm). The Caffarelli-Kohn-Nirenberg interpolation inequalities

w L 2p,γ (R d ) ≤ C β,γ,p ∇w ϑ L 2,β (R d ) w 1-ϑ L p+1,γ (R d ) (5)
with ϑ := are valid for all functions in the space obtained by completion of D(R d ) with respect to the norm • defined by w → w 2 = ∇w 2 L 2,β (R d ) + w 2 L p+1,γ (R d ) : see [4, Section 2.1] for more details. We shall take for granted once for all assumptions [START_REF] Bonforte | Global positivity estimates and Harnack inequalities for the fast diffusion equation[END_REF] on the parameters, even when it is not mentioned explicitly. However, the subcriticality condition p ≤ p will be assumed for global decay estimates, but not in the study of asymptotic decay estimates.

In [START_REF] Bonforte | Special fast diffusion with slow asymptotics: entropy method and flow on a Riemann manifold[END_REF], C β,γ,p is meant to be the best constant. In Part I of this study, [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF], we have showed that the equality case is achieved by w = B m-1/2 when p = 1/(2 m -1) if symmetry holds in [START_REF] Bonforte | Special fast diffusion with slow asymptotics: entropy method and flow on a Riemann manifold[END_REF], that is, if optimality is achieved among radial functions. However, we have proved in [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF] that symmetry breaking takes place if γ < 0 and β FS (γ) < β < d-2 d γ , with β FS (γ) := d -2 -(γd) 2 -4 (d -1) . [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF] In this case we have C β,γ,p := ∇w -ϑ L 2,β (R d ) w ϑ-1 L p+1,γ (R d ) w L 2p,γ (R d ) < C β,γ,p . On the contrary, according to [START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF], symmetry holds, so that C β,γ,p = C β,γ,p , if 0 ≤ γ < d , or γ < 0 and β ≤ β FS .

In this case, inequality [START_REF] Bonforte | Special fast diffusion with slow asymptotics: entropy method and flow on a Riemann manifold[END_REF] can be written as an entropy -entropy production inequality

(2 + β -γ) 2 F[v] ≤ m 1 -m I[v]
and, as a straightforward consequence (recall (4)), we formally deduce the convergence rate

F[v(t)] ≤ F[v(0)] e -2 (1-m) Λ t ∀ t ≥ 0 with Λ := (2+β-γ) 2 2 (1-m) . (8)
This connection is well known and goes back to [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF], where similar properties were investigated in the non-weighted case, namely for (β, γ) = (0, 0). For more details see [4, Proposition 1]. Conditions have to be given to make Estimate (8) rigorous.

In fact much more is known. Let us consider again the entropy -entropy production inequality

K(M ) F[v] ≤ I[v] ∀ v ∈ L 1,γ (R d ) such that v L 1,γ (R d ) = M , (9)
where K(M ) is the best constant, which may possibly take the value 0. With Λ(M ) := m 2 (1m) -2 K(M ), we formally deduce from (4) and [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] that

F[v(t)] ≤ F[v(0)] e -2 (1-m) Λ(M ) t ∀ t ≥ 0 . (10)
For brevity, this is what we shall call a global rate of decay because the dependence in v at time t = 0 is explicitly given by F[v(0)]. Let

m 1 := 2 d -2 -β -γ 2 (d -γ)
and denote by Λ 0,1 the lowest eigenvalue associated with non-radial eigenfuntions of the operator L defined by

L f := |x| γ B m-2 ∇ • |x| -β B ∇f ,
where B denotes the Barenblatt profile defined by

B(x) = C + |x| 2+β-γ 1 m-1 ∀ x ∈ R d (11)
for some C > 0. Of course we shall later take C = C(M ) if m ∈ (m c , 1). Based on variational methods, the following result has been proved in [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF]. We shall also give a short additional proof of (i) below in Section 4.2.

Theorem 1. Let [START_REF] Bonforte | Global positivity estimates and Harnack inequalities for the fast diffusion equation[END_REF] hold and m ∈ [m 1 , 1). With the above notations, we have:

(i) For any M > 0, if Λ(M ) = Λ then β = β FS (γ), (ii) If β > β FS (γ) then Λ 0,1 < Λ and Λ(M ) ∈ (0, Λ 0,1 ] for any M > 0, (iii) For any M > 0, if β < β FS (γ) and γ < 0, or if 0 ≤ γ < d, then Λ(M ) > Λ ,
where β FS (γ) is given by [START_REF] Caffarelli | First order interpolation inequalities with weights[END_REF].

We shall assume that the initial datum v 0 is sandwiched between two Barenblatt profiles: there exist two positive constants C 1 and C 2 such that

B 1 (x) := C 1 + |x| 2+β-γ 1 m-1 ≤ v 0 (x) ≤ C 2 + |x| 2+β-γ 1 m-1 =: B 2 (x) ∀ x ∈ R d .
(12) Condition ( 12) may look rather restrictive, but it is probably not, because it is expected that the condition is satisfied, for some positive t, by any solution with nonnegative initial datum having finite initial free energy, as it is the case when (β, γ) = (0, 0) and for m sufficiently close to 1: see for instance [START_REF] Bonforte | Global positivity estimates and Harnack inequalities for the fast diffusion equation[END_REF][START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF]. However, initial regularization effects of (2) are out of the scope of the present paper. If m is not close to 1, Condition [START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF] determines, to some extent, the basin of attraction of B. What we shall prove is the following result on global rates. Corollary 2. Let (6) hold and m ∈ [m 1 , 1). With the above notations, (10) holds if v is a solution to (2) with initial datum subject to [START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF].

We know that symmetry holds in (5) whenever β ≤ β FS (γ) and γ < 0, or if 0 ≤ γ < d. The restriction m ≥ m 1 comes from the (sub-)criticality condition p ≤ p in [START_REF] Bonforte | Special fast diffusion with slow asymptotics: entropy method and flow on a Riemann manifold[END_REF]. As in [START_REF] Blanchet | Hardy-Poincaré inequalities and applications to nonlinear diffusions[END_REF][START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], if one is interested only in the asymptotic rate of decay of F[v(t)] as t → ∞ (i.e., without requiring that the multiplicative constant is F[v(0)]), this restriction can be lifted and better estimates of the rates can be given using an appropriate linearization of the problem. Let us give some explanations in this regard.

Still at a formal level, we may consider a solution v = B (1 + ε B 1-m f ) to (2) and keep only the first order term in ε, as in [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF]. The corresponding linear evolution equation is

f t = (1 -m) L f .
Mass conservation (more precisely, relative mass conservation, see Sections 2.2 and 2.3), which is taken into account by requesting that

R d f B 2-m dx |x| γ = 0 ,
suggests to analyse the spectral gap of L considered as an operator acting on the space L 2 (R d , B 2-m |x| -γ dx). The reader interested in more details is invited to refer to [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF]Section 3.3]. Let us define

m * := d -4 -2 β + γ d -2 -β , α := 1 + β -γ 2 , δ := 1 1 -m , n := 2 d -γ β + 2 -γ , (13)
and pick the unique positive solution to η (η + n -2) = (d -1)/α 2 , which is given by

η = d-1 α 2 + n-2 2 2 -n-2 2 = 2 2+β-γ d -1 + d-2-β 2 2 -d-2-β 2+β-γ .
The following result can be deduced from [4, Lemma 8] (please note the discrepancy of a factor α 2 , which is due to the change of variables x → x |x| α-1 ). See in particular [4, Figure 4 and Appendix B] for details.

Proposition 3 (A Hardy-Poincaré-type inequality). Let d ≥ 2, m ∈ (0, 1) and (6) holds. For any f ∈ L 2 (R d , B 2-m |x| -γ dx), such that R d f B 2-m |x| -γ dx = 0 if m > m * , there holds R d |∇f | 2 B dx |x| β ≥ Λ R d |f | 2 B 2-m dx |x| γ .
The optimal constant is Λ = Λ ess if δ ≤ (n + 2)/2 and Λ = min {Λ ess , Λ 0,1 , Λ 1,0 } otherwise, with

Λ ess = 1 4 (n -2 -2 δ) 2 , Λ 0,1 = 2 δ η , Λ 1,0 = 2 (2 δ -n) .
Notice that the optimal constant Λ in the Hardy-Poincaré-type inequality is independent of C. In the range m ∈ (m c , 1), it is also independent of M since B as defined in [START_REF] Dolbeault | Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF] with C = C(M ) explicitly depends on M , according to the expression of C(M ) given in [4, Appendix A].

In Proposition 3, Λ ess , Λ 0,1 and Λ 1,0 , respectively, denote the infimum of the essential spectrum of L, the lowest positive eigenvalue associated with a non-radial eigenfunction, and the lowest positive eigenvalue associated with a radial eigenfunction. In practice, we have

Λ = Λ 1,0 if (n + 2)/2 ≤ δ ≤ n/(2 -η) and Λ = Λ 0,1 if δ > n/(2 -η), with (n -2)/2 < (n + 2)/2 < n/(2 -η) and Λ ess = 0 if and only if δ = (n -2)/2. See Figure 1.
Notice that m * as defined in [START_REF] Dolbeault | L q -functional inequalities and weighted porous media equations[END_REF] is the unique value of m for which, eventually, Λ ess = 0 and, as a consequence, for which there is no spectral gap. Notice that for some values of d, γ and β, the exponent m * takes nonpositive values. However our results are limited to m ∈ (0, 1) and in particular m > 0 will be assumed throughout this paper. Since δ = 1/(1m), we obtain precisely the value given by [START_REF] Dolbeault | L q -functional inequalities and weighted porous media equations[END_REF]

. If m > m * then B 2 -B 1 is in L 1,γ (R d ),
where B 1 and B 2 are defined as in [START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF]. This is not anymore true if m ≤ m * . In that case we shall consider L 1,γ -perturbations of B as defined in [START_REF] Dolbeault | Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF], for some constant

C ∈ [C 2 , C 1 ]. If m > m * then the condition R d (v 0 -B) |x| -γ dx = 0
uniquely determines C = C(M ). We shall refer to these conditions as the relative mass condition: see Assumptions (H1) and (H2) in Section 2.1 for further details.

We are now able to state the main results of this paper.

Theorem 4. Let (6) hold and m ∈ (0, 1), with m = m * . Under the relative mass condition and with same notations as in Proposition 3, if v solves (2) subject to [START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF], then there exists a positive constant C such that

F[v(t)] ≤ C e -2 (1-m) Λ t ∀ t ≥ 0 . 0 Λ 0,1 Λ 1,0 Λ ess Essential spectrum δ δ4 δ1 δ5 δ2 Λ0,1 Λ1,0 Λess Essential spectrum δ4 δ5:= n 2-η Figure 1.
The spectrum of L as a function of δ = 1 1-m , with n = 5. The essential spectrum corresponds to the grey area, and its bottom is determined by the parabola δ → Λ ess (δ). The two eigenvalues Λ 0,1 and Λ 1,0 are given by the plain, half-lines, away from the essential spectrum. Note that solutions of the eigenvalue problem exist for any value of δ but may not be in the domain of the operator or below the essential spectrum and are then represented as dotted half-lines. See [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF]Appendix B] for a discussion of the values of δ 1 , δ 2 ,... δ 5 . The right figure is an enlargement of the left one. This configuration is not generic: see [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF]Fig. 5] for other cases.

As in [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], this result itself relies on a result of relative uniform convergence which is the key estimate to relate the free energy to the spectrum of L. Before stating the latter, let us define

ζ := 1 -1 -2-m (1-m) q 1 -2-m 1-m θ where θ := (1-m) (2+β-γ) (1-m) (2+β)+2+β-γ
and observe that in view of hypotheses [START_REF] Bonforte | Global positivity estimates and Harnack inequalities for the fast diffusion equation[END_REF] and of the change of variables [START_REF] Dolbeault | L q -functional inequalities and weighted porous media equations[END_REF], θ is in the range 0 < θ < 1-m 2-m < 1. Theorem 5. Under the assumptions of Theorem 4, there exist positive constants K and t 0 such that, for all q ∈ 2-m 1-m , ∞ , the function w = v/B satisfies

w(t) -1 L q,γ (R d ) ≤ K e -2 (1-m) 2 2-m Λ ζ (t-t0) ∀ t ≥ t 0 (14)
in the case γ ∈ (0, d), and

w(t) -1 L q,γ (R d ) ≤ K e -2 (1-m) 2 2-m Λ (t-t0) ∀ t ≥ t 0 in the case γ ≤ 0.
We point out that Estimate ( 14) yields an improvement of a similar result, namely [2, Theorem 3], in the non-weighted case (β, γ) = (0, 0). We shall comment more on the rates of convergence provided by Theorem 5 in Section 4.3.

The proof of Theorem 5 partially relies on uniform Hölder-regularity estimates for bounded solutions to a linearized version of Equation [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF]. In view of possible degeneracies or singularities of the weights |x| -γ and |x| -β at the origin, such results do not follow from standard parabolic theory and therefore have to be proved separately. We devote an Appendix to these issues, where we give sketches of proofs. These are based on a strategy developed for similar equations by [START_REF] Chiarenza | A remark on a Harnack inequality for degenerate parabolic equations[END_REF].

Using refinements that will be discussed in Section 4.1, we can also prove convergence results in L 1,γ norms. For this purpose, we need to restrict the range of m to ( m 1 , 1), where m 1 is the smallest number such that R d |x| 2+β-γ B |x| -γ dx is finite for all m ∈ ( m 1 , 1), that is

m 1 := d -γ d + 2 + β -2 γ .
Let us introduce the rescaled function

v µ (t, x) := µ γ-d v(t, x/µ) ∀ (t, x) ∈ R + × R d , ∀ µ > 0 .
Theorem 6. Under the assumptions of Theorem 4 with in addition m ∈ ( m 1 , 1), there exists a monotone, positive function t → µ(t) with lim t→+∞ µ(t) = 1 such that

lim sup t→∞ e (1-m) min{Λess,Λ0,1} t v µ(t) (t) -B 1 + |x| 2+β-γ L 1,γ (R d ) < ∞ .
This result is an improvement in the spirit of [START_REF] Dolbeault | Fast diffusion equations: matching large time asymptotics by relative entropy methods[END_REF] in the non-weighted case (β, γ) = (0, 0). As it appears in Proposition 3 and Theorems 4 and 5, Λ is smaller than min{Λ ess , Λ 0,1 } only under conditions that are discussed in [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF]Appendix B]. Also notice that, by undoing the self-similar change of variables outlined in Section 2.3, it is possible to give algebraic rates of convergence for the original solutions to (1), as in [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF].

Let us conclude this introduction by a few bibliographical references. In the case without weights, we primarily refer to [START_REF] Blanchet | Hardy-Poincaré inequalities and applications to nonlinear diffusions[END_REF][START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] and references therein. The special case corresponding to δ = (n-2)/2 has been treated in [START_REF] Bonforte | Special fast diffusion with slow asymptotics: entropy method and flow on a Riemann manifold[END_REF]. Still in the non-weighted case, improvements have been obtained more recently in [START_REF] Dolbeault | Fast diffusion equations: matching large time asymptotics by relative entropy methods[END_REF][START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF][START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF][START_REF] Dolbeault | Best matching Barenblatt profiles are delayed[END_REF][START_REF] Dolbeault | Best matching Barenblatt profiles are delayed[END_REF][START_REF] Dolbeault | Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays[END_REF][START_REF] Dolbeault | Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities[END_REF] using refinements of relative entropy methods, and in [START_REF] Denzler | Higher-order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach[END_REF] using a detailed analysis of fast and slow variables and of the invariant manifolds. These papers are anyway limited to the choice (β, γ) = (0, 0). More references can be found therein.

As for problems with power law weights, we shall refer to [START_REF] Reyes | The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions[END_REF][START_REF] Reyes | Long time behavior for the inhomogeneous PME in a medium with slowly decaying density[END_REF][START_REF] Kamin | Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density[END_REF] for approaches based on comparison techniques in the case β = 0 and for the porous media equation. The papers [START_REF] Nieto | Asymptotic behavior of the solutions of the inhomogeneous porous medium equation with critical vanishing density[END_REF][START_REF] Iagar | Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density[END_REF] deal with the critical power |x| -2 , where asymptotics is more subtle. A detailed long-time analysis has been carried out in [START_REF] Grillo | On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density[END_REF] for the fractional porous media equation with a weight. Diffusion equations of porous media type with two weights (i.e. weights having the same role as |x| -γ and |x| -β here) have been investigated, e.g., in [START_REF] Dolbeault | L q -functional inequalities and weighted porous media equations[END_REF][START_REF] Grillo | Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincaré inequalities[END_REF], where well-posedness issues as well as smoothing effects and asymptotic estimates are discussed in rather general weighted frameworks by means of functional inequalities. In the fast diffusion regime, convergence in relative error to a separable profile for radial solutions on the hyperbolic space has been proved by [START_REF] Grillo | Radial fast diffusion on the hyperbolic space[END_REF], through pure barrier methods. Note that, in radial coordinates, the Laplace-Beltrami operator is in fact a two-weight Laplacian. The corresponding analysis for the porous medium equation (for general solutions) has then been carried out in [START_REF] Vázquez | Fundamental solution and long time behavior of the porous medium equation in hyperbolic space[END_REF].

A detailed justification of the introduction of weights and especially power law weights in case of porous media and fast diffusion equations can be found in [START_REF] Kamin | Propagation of thermal waves in an inhomogeneous medium[END_REF][START_REF] Rosenau | Nonlinear diffusion in a finite mass medium[END_REF]. In [START_REF] Dolbeault | Weighted interpolation inequalities: a perturbation approach[END_REF], for β = 0 and γ > 0 small enough, symmetry of optimal functions in the Caffarelli-Kohn-Nirenberg inequalities (5) is proved to hold. Notice however that the inequalities are then more of Hardy-Sobolev type than of Caffarelli-Kohn-Nirenberg because only one weight is involved. The other case of symmetry in Caffarelli-Kohn-Nirenberg inequalities, which is now fully understood, is the one corresponding to the threshold case p = p , which has been recently solved in [START_REF] Dolbeault | Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF]. Remarkably the proof relies on the very same flow (1) and an approach based on the Bakry-Emery Γ 2 method. The reader interested in further considerations on Caffarelli-Kohn-Nirenberg inequalities, Γ 2 computations and rigidity results in nonlinear elliptic problems on compact and non-compact manifolds is invited to refer to this paper for a more complete review of the literature in this direction.

The paper is organized as follows. Properties of self-similar solutions, an existence result, a comparison result, the conservation of the relative mass, results on the relative entropy and the rewriting of (1) in relative variables after a self-similar change of variables have been collected in Section 2. These results are adapted form the case (β, γ) = (0, 0). Section 3 is devoted to regularity issues and to the relative uniform convergence, that is, the uniform convergence of the quotient of the solution in self-similar variables by the Barenblatt profile: this at the core of our results and it is also where our paper differs from the case (β, γ) = (0, 0). There we prove Theorems 4 and 5. Because of the weights, the Hölder regularity at the origin is an issue. It relies on a technical result, based on an adaptation of [START_REF] Chiarenza | A remark on a Harnack inequality for degenerate parabolic equations[END_REF]: the proof is given in an Appendix. Some additional results, including the proof of Theorem 6 and some comments have been collected in Section 4.

Throughout this paper, B ρ denotes the centered ball of radius ρ, that is,

B ρ := {x ∈ R d : |x| < ρ}.
2. Self-similar variables, relative entropy and large time asymptotics.

2.1.

The self-similar solutions. In order to avoid confusion between original variables and rescaled variables, let us rewrite (1) as

|y| -γ u τ + ∇ • |y| -β u ∇u m-1 = 0 ∀ (τ, y) ∈ R + × R d .
The whole family of explicit self-similar solutions of Barenblatt type is given by

U C,T (τ, y) := R(τ ) γ-d C + |y|/R(τ ) 2+β-γ 1/(1-m) ∀ (τ, y) ∈ R + × R d , (15)
where C, T > 0 are free parameters and R(τ ) is defined by

dR dτ = R 1-(d-γ) (m-mc) , R(0) = [(d -γ) |m -m c | T ] 1 (d-γ) (m-mc ) (16)
if m = m c , with m c as above, namely

m c := d -2 -β d -γ ∈ (0, 1) .
In the special case m = m c we shall replace the initial condition with R(0

) = e T , for T ∈ R. More explicitly, R(τ ) =          [(d -γ) (m -m c ) (T + τ )] 1 (d-γ) (m-mc ) if m ∈ (m c , 1) , ∀ τ ≥ 0 , e T +τ if m = m c , ∀ τ ≥ 0 , [(d -γ) (m c -m) (T -τ )] - 1 (d-γ) (mc -m) if m ∈ (0, m c ) , ∀ τ ∈ [0, T ) .
If m ≥ m c Barenblatt-type solutions are positive for all τ > 0. If m < m c these solutions extinguish at τ = T .

As already mentioned in Section 1, we shall require that the initial datum u(0) = u 0 is trapped between two Barenblatt profiles. More precisely:

(H1) There exist positive constants T and

C 1 > C 2 such that U C1,T (0, y) ≤ u 0 (y) ≤ U C2,T (0, y) ∀ y ∈ R d . (H2) There exist C ∈ [C 2 , C 1 ] and f ∈ L 1,γ (R d ) such that u 0 (y) = U C,T (0, y) + f (y) ∀ y ∈ R d .
If m < m c solutions with initial datum as above extinguish at t = T < ∞ as we shall deduce from the comparison principle (see Corollary 9 and related comments below). Such solutions do do not belong to L 1,γ (R d ). On the other hand, if m ≥ m c solutions are positive at all τ > 0. They belong to L

1,γ (R d ) if in addition m > m c . If m > m * , with m * := d -4 -2 β + γ d -2 -β < m c ,
Assumption (H2) is in fact a consequence of (H1). Indeed in such a range Barenblatt solutions may not be in L 1,γ (R d ) but the difference of two Barenblatt profiles still belongs to L 1,γ (R d ). On the contrary, if m ≤ m * then (H2) induces an additional restriction.

2.2. Existence, comparison and conservation of relative mass. In agreement with [START_REF] Herrero | The Cauchy problem for ut = ∆u m when 0 < m < 1[END_REF], we provide the following definition of a weak solution.

Definition. For a nonnegative u 0 ∈ L ∞ (R d ), by a solution to (1) we mean a nonnegative function u ∈ C([0, ∞); L 1,γ loc (R d )) ∩ L ∞ (R + × R d ) satisfying - R + R d u ϕ τ |y| -γ dy dτ = 1 -m m R + R d u m ∇ • |y| -β ∇ϕ dy dτ for all ϕ ∈ D(R + × R d ) and lim τ →0 + u(τ ) = u 0 .
In [START_REF] Herrero | The Cauchy problem for ut = ∆u m when 0 < m < 1[END_REF], (β, γ) = (0, 0) and we point out that it is only required that

u 0 ∈ L 1 loc (R d ) and u ∈ C([0, ∞); L 1 loc (R d ))
. However, because of the weight |y| -β , a priori the equation may not make sense, since in general u m ∈ L 1,β loc (R d ). Hence, for simplicity, we also assume that initial data and solutions are globally bounded, as in the sequel we shall only deal with this kind of solutions.

Proposition 7 (Existence). Assume that m ∈ (0, 1). For any nonnegative u 0 ∈ L ∞ (R d ) there exists a solution to (1) in the sense of the above definition.

Proof. We refer the reader to the proof of [25, Theorem 2.1]: minor changes have to be implemented in order to adapt it to our weighted context. The basic idea consists in approximating the initial datum, e.g., with the sequence

u 0n := φ n u 0 ∈ L 1,γ (R d ) ∩ L ∞ (R d ) where φ n = φ(•/n) and φ is a smooth truncation function such that 0 ≤ φ ≤ 1, φ(x) = 0 if |x| > 2 and φ(x) = 1 if |x| ≤ 1.
The corresponding sequence of solutions u n is well defined in view of standard L 1,γ theory (there is no additional difficulty due to the weights compared to the standard theory as exposed in [START_REF] Vázquez | The porous medium equation[END_REF]). One can then pass to the limit on such a sequence by exploiting local L 1,γ estimates (as in [START_REF] Herrero | The Cauchy problem for ut = ∆u m when 0 < m < 1[END_REF]Lemma 3.1]) along with the global bound u n (τ ) ∞ ≤ u 0n ∞ , valid for all τ > 0.

Proposition 8 (L 1,γ -contraction). Assume that m ∈ (0, 1) and let u 01 , u 02 ∈ L ∞ (R d ) be any two nonnegative initial data with corresponding solutions u 1 , u 2 to (1), that are constructed via the approximation scheme of the proof of Proposition 7. Then

R d [u 1 ( τ , y) -u 2 ( τ , y)] + dy |y| γ ≤ R d [u 1 (τ, y) -u 2 (τ, y)] + dy |y| γ ∀ τ ≥ τ ≥ 0 .
Proof. The inequality holds for the approximate solutions u 1,n and u 2,n still as a consequence of the standard L 1,γ theory. Hence, the assertion just follows by taking limits as n → ∞.

Proposition 8 trivially implies the following key comparison result.

Corollary 9 (Comparison principle). Under the same hypotheses as in Proposition 8, if u 01 ≤ u 02 , then u 1 (τ ) ≤ u 2 (τ ) for all τ ≥ 0.

As the reader may note, we do not claim that we have a comparison principle (and hence a uniqueness result) for any solutions in the sense of Definition 2.2, but only for those obtained as limits of L 1,γ approximations. Nevertheless, in the sequel by solution we shall tacitly mean the one constructed as in the proof of Proposition 7, for which comparison holds. Since we consider initial data satisfying (H1), in order to conclude that the corresponding solutions are trapped between Barenblatt profiles at any time, one has to check that the self-similar solution given by ( 15) can also be obtained as a limit of L 1,γ approximate solutions. This is a standard fact given the explicit profile of U C,T .

Mass conservation is used in the range m > m c to determine the parameter C = C(M ) which characterizes the Barenblatt profile U C,T having the same mass as u.

In the range m ≤ m c we can still prove that the quantity

R d [u(τ, y) -U C,T (τ, y)] |y| -γ dy ,
which we shall refer to as relative mass, is conserved at any τ > 0, even if

U C,T (τ ) ∈ L 1,γ (R d ).
Proposition 10 (Conservation of relative mass). Assume that m ∈ (0, 1) and consider a solution u of (1) with initial datum u 0 satisfying (H1)-(H2). Then

R d [u(τ, y) -U C,T (τ, y)] dy |y| γ = R d [u 0 (y) -U C,T (0, y)] dy |y| γ ∀ τ ≥ 0 .
Proof. We proceed along the lines of the proof of

[2, Proposition 1]. That is, let φ be a C 2 (R + ) function such that 0 ≤ φ ≤ 1, φ(x) = 0 if |x| > 2 and φ(x) = 1 if |x| ≤ 1. For any λ > 0, set φ λ (y) := φ(|y|/λ). Then d dτ R d [u(τ, t) -U C,T (τ, y)] φ λ dy |y| γ = 1 -m m B 2λ \B λ u m (τ, y) -U m C,T (τ, y) ∇ • |y| -β ∇φ λ dy ≤ (1 -m) B 2λ \B λ U m-1 C1,T |u -U C,T | |y| -β |∆φ λ | + β |y| -β-1 |∇φ λ | dy .
As λ → ∞, we observe that U m-1 C1,T , |∆φ λ | and |∇φ λ | behave like λ 2+β-γ , λ -2 and λ -1 , respectively, in the region B 2λ \ B λ . In particular,

U m-1 C1,T |y| -β |∆φ λ | + β |y| -β-1 |∇φ λ | ≤ c |y| -γ ∀ y ∈ B 2λ \ B λ
for a suitable c > 0 independent of λ. Hence, for all τ 2 > τ 1 ≥ 0 we deduce that

R d [u(τ 2 , y) -U C,T (τ 2 , y)] φ λ dy |y| γ - R d [u(τ 1 , y) -U C,T (τ 1 , y)] φ λ dy |y| γ ≤ (1 -m) c τ2 τ1 B 2λ \B λ |u(τ, y) -U C,T (τ, y)| dy |y| γ dτ .
The L 1,γ -contraction of Proposition 8 ensures that the r.h.s. vanishes as λ → ∞.

Actually, since a priori u τ does not exist as a function, we have to use a test function φ λ that depends on time and whose time derivative approximates the difference between two Dirac deltas at times τ 2 and τ 1 . However, this is a standard technicality, which we omitted in order to make the proof more readable.

2.3. Self-similar variables: a nonlinear Fokker-Plank equation. As already discussed in [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF] and outlined in Section 1, it is convenient to analyse the asymptotic behaviour of solutions to (1) whose initial data comply with (H1)-(H2) by means of a suitable time-space change of variables that makes Barenblatt profiles stationary.

Let us rescale the function u according to

u(τ, y) = R(τ ) γ-d v log R(τ ) R(0) , y R(τ ) , t = log R(τ ) R(0) , x = y R(τ ) ,
where R is defined by [START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF]. Similarly, the Barenblatt solution U C,T is transformed into B as defined by [START_REF] Dolbeault | Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF]. Straightforward computations show that v solves the nonlinear, weighted Fokker-Plank equation

|x| -γ v t = -∇ • |x| -β v ∇ v m-1 -B m-1 ∀ (t, x) ∈ R + × R d . ( 17
)
In terms of the initial datum v 0 (x) = R(0) d-γ u 0 (R(0) x), conditions (H1)-(H2) can be rewritten as follows:

(H1') There exist positive constants

C 1 > C 2 such that B 1 (x) = C 1 + |x| 2+β-γ -1 1-m ≤ v 0 (x) ≤ C 2 + |x| 2+β-γ -1 1-m = B 2 (x) for all x ∈ R d . (H2') There exist C ∈ [C 2 , C 1 ] and f ∈ L 1,γ (R d ) such that v 0 (x) = B(x) + f (x) ∀ x ∈ R d .
Assumption (H1') is nothing else than [START_REF] Dolbeault | Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities[END_REF]. Note that, for greater readability, in (H2') we have replaced R(0

) d-γ f (R(0) x) with f (x) again. As mentioned in Section 2.1, if m > m * the difference B 2 -B 1 is in L 1,γ (R d ), so that (H2') is implied by (H1') and the map C → R d (v 0 -B) |x| -γ dx is continuous, monotone increasing and changes sign in [C 2 , C 1 ]. Hence, in this case there exists a unique C ∈ [C 2 , C 1 ] such that R d (v 0 -B) dx |x| γ = 0 .
It is clear that, as a consequence of Proposition 10, under assumptions (H1')-(H2') the relative mass of v is also conserved, that is

R d [v(t, x) -B(x)] dx |x| γ = R d (v 0 -B) dx |x| γ ∀ t > 0 provided R d |v 0 -B| |x| -γ dx is finite for some C > 0. If m > m * , we deduce that R d [v(t, x) -B(x)] dx |x| γ = 0 ∀ t > 0 .
On the other hand, if m ≤ m * we cannot ensure that this identity still holds, but, nevertheless, the conservation of relative mass is still true and reads

R d [v(t, x) -B(x)] dx |x| γ = R d f dx |x| γ ∀ t > 0
, where the r.h.s. does not necessarily takes the value 0.

2.4. The relative error: a nonlinear Ornstein-Uhlenbeck equation. Consider a solution v of ( 17) corresponding to an initial datum that satisfies (H1')-(H2'). As in [2, Section 2.3], let us introduce the ratio

w(t, x) := v(t, x) B(x) ∀ (t, x) ∈ R + × R d .
The difference w -1 is usually referred to as relative error between v and B. In view of [START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF], it is straightforward to check that w is a solution to

   |x| -γ w t = -1 B ∇ • |x| -β B w ∇ (w m-1 -1) B m-1 in R + × R d , w(0, •) = w 0 := v 0 B in R d , (18)
which can be seen as a nonlinear, weighted equation of Ornstein-Uhlenbeck type.

Let us also define the quantities

W 1 := inf x∈R d B 1 (x) B(x) ≤ sup x∈R d B 2 (x) B(x) =: W 2 .
A straightforward calculation yields

0 < W 1 = C C 1 1 1-m ≤ 1 ≤ C C 2 1 1-m = W 2 < ∞ .
In terms of w 0 , assumptions (H1') and (H2') can in turn be rewritten as follows:

(H1") There exist positive constants

C 1 > C 2 and C ∈ [C 2 , C 1 ] such that W 1 ≤ B 1 (x) B(x) ≤ w 0 (x) ≤ B 2 (x) B(x) ≤ W 2 ∀ x ∈ R d . (H2") There exists f ∈ L 1,γ (R d ) such that w 0 (x) = 1 + f (x) B(x) ∀ x ∈ R d .
Note that, as a consequence of the L 1,γ -contraction estimate and the comparison principle (Proposition 8 and Corollary 9), assumptions (H1")-(H2") are satisfied by a solution w(t) of ( 18) at any t > 0 if they are satisfied by w 0 , for a suitable f depending also on t (clearly the same holds for v, with respect to (H1')-(H2')).

3. Regularity, relative uniform convergence and asymptotic rates. The goal of this section is to show that the relative error w(t) -1 converges to zero uniformly as t → ∞. Then in Section 3.2, by taking advantage of this result, we shall prove that such convergence occurs with explicit exponential rates. Lemma 11. Assume that m ∈ (0, 1). Let w be the solution of (18) corresponding to an initial datum w 0 satisfying (H1")-(H2"). Then there exist ν ∈ (0, 1) and a positive constant K > 0, depending on d, m, β, γ, C, C 1 , C 2 such that:

∇v(t) L ∞ (B 2λ \B λ ) ≤ Q 1 λ 2+β-γ 1-m +1 ∀ t ≥ 1 , ∀ λ > 1 , (19)
sup t≥1 w C k ((t,t+1)×B c ε ) < ∞ ∀ k ∈ N , ∀ ε > 0 , (20)
sup t≥1 w(t) C ν (R d ) < ∞ , (21)
sup

τ ≥t |w(τ ) -1| C ν (R d ) ≤ K sup τ ≥t w(τ ) -1 L ∞ (R d ) ∀ t ≥ 1 , ( 22
)
where v is the solution of (17) corresponding to the initial datum v 0 = w 0 B.

Proof. For all λ > 1, let us consider the following rescaling:

v λ (t, x) := λ 2+β-γ 1-m v(t, λ x) ∀ (t, x) ∈ R + × R d .
It is straightforward to check that v λ satisfies the same equation as v, with initial datum (v 0 ) λ . In particular, since (v 0 ) λ is bounded and bounded away from zero in B 2 \ B ε/2 independently of λ (consequence of (H1')), in view of standard parabolic regularity there holds

|v λ | C k ((t,t+1)×(B1\Bε)) ≤ Q k ∀ t ≥ 1
for all k ∈ N and some Q k > 0 depending only on d, m, β, γ, C, C 1 , C 2 and ε, but independent of λ. By undoing the scaling, this is equivalent to

|v| C k x ((t,t+1)×(B λ \B ελ )) ≤ Q k λ 2+β-γ 1-m +k , |v| C k τ ((t,t+1)×(B λ \B ελ )) ≤ Q k λ 2+β-γ 1-m ∀ t ≥ 1 ,
where by C k x and C k τ we mean partial derivatives restricted to space and time, respectively. As a special case, this proves [START_REF] Dolbeault | Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays[END_REF] and [START_REF] Dolbeault | Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities[END_REF] upon observing that

B -1 L ∞ (B λ \B ελ ) ≈ λ 2+β-γ 1-m , |B| C k (B λ \B ελ ) ≈ λ -2+β-γ 1-m -k . ( 23
)
As for proving [START_REF] Giusti | Direct methods in the calculus of variations[END_REF], it is enough to notice that, for some ν ∈ (0, 1) and another constant Q ν > 0 depending on the same quantities as Q k , with the exception of ε, the estimate

|v(t)| C ν (B 1/2 ) ≤ Q ν ∀ t ≥ 1
follows from the regularity results of the Appendix. Indeed, since v is bounded and bounded away from zero in R + × B 1 , we can apply Corollary 25 with the choices

a(t, x) = (1 -m) v m-1 (t, x) , B(t, x) = - 2 + β -γ 1 -m x |x| -γ v 1-m (t, x) .
As for [START_REF] Grillo | Radial fast diffusion on the hyperbolic space[END_REF], let z(t, x) := v(t, x) -B(x). Straightforward computations show that the equation solved by z reads

|x| -γ z t = ∇ • |x| -β a(t, x) (∇z + B(t, x) z) ,
with the same function a as above and

B(t, x) = v 1-m (t, x) v m-1 (t, x) -B m-1 (x) v(t, x) -B(x) -B m-2 (x) ∇B(x) .
We are therefore again in position to use Corollary 25 to get

|z(t)| C ν (B 2 -1/α ) ≤ K z L ∞ ((t+2,t+3)×B 2 2/α ) ∀ t ≥ 1 for some K > 0 depending on d, m, β, γ, C, C 1 , C 2 .
From here on K will denote a general positive constant, which may change from line to line. Corollary 25 holds with inessential modifications if one replaces balls with annuli. By performing scalings, we deduce that

|z(t)| C ν (B λ \B λ/2 ) ≤ K λ -ν z L ∞ ((t+2,t+3)×(B 8λ \B 2λ )) ∀ t ≥ 1 .
By standard computations and ( 23), which holds even if k is not an integer, and the identity w(t) -1 = z(t)/B, we obtain

|w(t) -1| C ν (B λ \B λ/2 ) ≤ B -1 L ∞ (B λ \B λ/2 ) |z(t)| C ν (B λ \B λ/2 ) + B -1 2 L ∞ (B λ \B λ/2 ) |B| C ν (B λ \B λ/2 ) z(t) L ∞ (B λ \B λ/2 ) ≤ K λ 2+β-γ 1-m -ν z L ∞ ((t+2,t+3)×(B 8λ \B 2λ )) + z(t) L ∞ (B λ \B λ/2 ) ≤ K λ -ν w -1 L ∞ ((t,t+3)×(B 8λ \B 2λ ))
.

By taking λ = 2 j/α , with j ≥ -1 integer, we conclude the proof of ( 22).

Remark 12. It is important to point out that the applicability of the results of the Appendix with coefficients a and B as in the proof of Lemma 11 relies on the fact that, under assumptions (H1')-(H2'), v is locally bounded and bounded away from zero. In other words, we do not claim that we have a Harnack inequality for general solutions to the degenerate/singular Equation (2).

b) The relative free energy and Fisher information. By proceeding along the lines of [2, Section 2.5], we redefine the relative free energy functional as

F[w] = 1 m -1 R d [w m -1 -m (w -1)] B m
|x| γ dx , with a slight abuse of notations in the sense that we consider it as a functional acting on w = v/B. Again, if we formally derive F[w(t)] with respect to t along the flow [START_REF] Dolbeault | Best matching Barenblatt profiles are delayed[END_REF] we obtain

d dt F[w(t)] = - m 1 -m I[w(t)] , ( 24
)
where I is the relative Fisher information, redefined in terms of w as

I[w] = R d w ∇ w m-1 -1 B m-1 2 B dx |x| β .
However, the rigorous justification of ( 24) is not straightforward, and to this end we need to take advantage of the global regularity estimates provided in Section 3.1.

Proposition 13 (Entropy-entropy production identity). Assume that m ∈ (0, 1). If w is a solution of (18) corresponding to an initial datum w 0 satisfying assumptions (H1")-(H2"), then the free energy F[w(t)] is finite for all t ≥ 0 and identity (24) holds.

Proof. We proceed through three steps, following the lines of proof of [2, Proposition 2]. We skip the proof of the fact that F[w(t)] is finite, since it goes exactly as in [2, Lemma 4], with inessential modifications. For the sake of greater readability we shall omit time-dependence, at least when this does not compromise comprehension.

• Step 1. Consider the same cut-off function φ λ as in the proof of Proposition 10, with λ > 1. Then, by using [START_REF] Dolbeault | Best matching Barenblatt profiles are delayed[END_REF], the identity w B = v and integrating by parts, we obtain

- d dt 1 1 -m R d [w m -1 -m (w -1)] φ λ B m |x| γ dx = - m 1 -m R d w ∇ v m-1 -B m-1 • ∇ (v m-1 -B m-1 ) φ λ B dx |x| β = - m 1 -m R d w ∇ v m-1 -B m-1 2 φ λ B dx |x| β + m 2 (1 -m) R(λ)
where

R(λ) := - R d ∇ (v m-1 -B m-1 ) 2 • ∇φ λ v dx |x| β = B 2λ \B λ v m-1 -B m-1 2 ∇v • ∇φ λ + v ∆φ λ -β v x |x| 2 • ∇φ λ dx |x| β .
We have

|R(λ)| (1 -m) 2 ≤ B 2λ \B λ |v -B| 2 B 2 (m-2) 1 |∇v| |∇φ λ | + v |∆φ λ | + β |x| v |∇φ λ | dx |x| β ≤ c B 2λ \B λ |v -B| dx |x| γ
where, in the last step, we used the inequality

|v -B| B 2 (m-2) 1 |∇v| |∇φ λ | + v |∆φ λ | + β |x| v |∇φ λ | 1 |x| β ≤ c |x| γ ∀ x ∈ B 2λ \ B λ (25
) for some c > 0, independent of λ > 1. We shall establish (25) in Step 2. By assumptions (H1')-(H2') and the L 1,γ -contraction principle, the difference v -B is in L 1,γ (R d ), so that lim λ→∞ R(λ) = 0 and the proof is completed by passing to the limit as λ → ∞. • Step 2. Recalling [START_REF] Dolbeault | Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays[END_REF], we know that

∇v(t) L ∞ (B 2λ \B λ ) ≤ Q 1 λ 2+β-γ 1-m +1 ∀ t ≥ 1 .
Moreover, since v is trapped between two Barenblatt profiles, we have

v(t) L ∞ (B 2λ \B λ ) ≤ Q 0 λ 2+β-γ 1-m , v(t) -B L ∞ (B 2λ \B λ ) ≤ Q 2 λ (2+β-γ) (2-m) 1-m .
The estimates hold for suitable positive constants Q 0 , Q 1 and Q 2 which are all independent of λ, t > 1. As for φ λ , by construction we have that

∇φ λ L ∞ (B 2λ \B λ ) ≤ c 1 λ , ∆φ λ L ∞ (B 2λ \B λ ) ≤ c 2 λ 2
, for some c 1 , c 2 > 0 which are also independent of λ. Estimate (25) readily follows.

• Step 3. It remains to take care of the origin. In principle solutions are only Hölder regular (see the Appendix). Nevertheless, since v is uniformly bounded and locally bounded away from zero, standard energy estimates (see again [START_REF] Vázquez | The porous medium equation[END_REF] as a general reference) ensure, e.g., that the quantities |x| γ dt are finite for all t 1 , t 2 > 0, which is enough in order to give sense to [START_REF] Grillo | On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density[END_REF] at least in a L 1 loc (R + ) sense.

c) Uniform convergence in relative error. By mimicking the proofs of [2, Lemma 5 and Corollary 1], we can show that the rescaled solution v converges to B uniformly in the strong sense of the relative error.

Proposition 14 (Convergence in relative error without rates). Assume that m ∈ (0, 1). If w is a solution of (18) corresponding to an initial datum w 0 satisfying assumptions (H1")-(H2"), then

lim t→∞ w(t) -1 L ∞ (R d ) = 0 .
Proof. For all τ > 0, we set w τ (t, x) := w(t + τ, x). In view of ( 20), there exists a sequence τ n → ∞ such that w τn converges locally uniformly in (1,

∞) × (R d \ {0}) to some w ∞ ∈ L ∞ ((1, ∞) × R d ).
Moreover, by Corollary 9 we deduce that

0 < W 1 ≤ B 1 (x) B(x) ≤ w ∞ (t, x) ≤ B 2 (x) B(x) ≤ W 2 < ∞ ∀ (t, x) ∈ (0, ∞) × R d . (26)
Thanks to Proposition 13, there holds

F[w(τ n + 1)] -F[w(τ n + 2)] = τn+2 τn+1 I[w(t)] dt = 2 1 I[w(t + τ n )] dt ≥ 0 .
Since F[w(s n + 2)] is bounded from below (as a consequence of (H1")-(H2"), see again [2, Lemma 4]), we infer that I[w τn (t)] converges to zero in L 1 ((1, 2)) as n → ∞, that is,

lim n→∞ 2 1 R d w τn (t, x) ∇ w m-1 τn (t, x) -1 B m-1 2 B dx |x| β dt = 0 .
By Fatou's lemma, this implies

2 1 R d lim inf n→∞ ∇ w m-1 τn (t, x) -1 B m-1 2 w ∞ (t, x) B dx |x| β dt = 0 .
Still as a consequence of ( 20) and ( 26) we have that

0 ≡ lim inf n→∞ ∇ w m-1 τn -1 B m-1 = ∇ w m-1 ∞ -1 B m-1 a.e. (1, 2) × R d .
This means that the function w m-1

∞ -1 B m-1 is constant, hence w ∞ (t, x) = 1 + c(t) B 1-m (x) -1 1-m ∀ (t, x) ∈ (1, 2) × R d .
It is readily seen that the only possibility is c ≡ 1. Indeed, if m > m * this is due to the conservation of relative mass (Proposition 10), while in the case m ≤ m * it is a consequence of the L 1,γ -contraction principle (Proposition 8). Since we can repeat the same argument as above, up to subsequences, along any sequence τ n → ∞, in fact we have shown that

lim t→∞ w(t) ≡ 1 in L ∞ loc (R d ) .
In order to obtain the global uniform convergence, it is enough to recall [START_REF] Iagar | Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density[END_REF] and note that by dominated convergence we have lim t→∞ w(t) -1 L p (R d ) = 0 for all p > d/(2 + βγ): the global C ν estimate, [START_REF] Giusti | Direct methods in the calculus of variations[END_REF], and a standard interpolation like [2, Proof of Theorem 1] allow us to conclude.

3.2.

Hardy-Poincaré inequalities: convergence with rates. As in [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], if m = m * , sharp rates of convergence towards the Barenblatt profile B are related to the optimal constant Λ > 0 of the Hardy-Poincaré-type inequality

R d |∇f | 2 B dx |x| β ≥ Λ R d |f | 2 B 2-m dx |x| γ ( 27
)
for any function f ∈ C ∞ c (R d ) such that, additionally, R d f B 2-m dx |x| γ = 0 whenever R d B 2-m dx |x| γ is finite, that is, for m > m * .
The explicit value of Λ has been computed explicitly in [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF], and is provided in Proposition 3.

• Weighted linearization. In order to better understand the asymptotic behaviour of the solutions at hand, let us outline our strategy. The idea, as in [2, Section 3.3], is to linearize the equation of the relative error [START_REF] Dolbeault | Best matching Barenblatt profiles are delayed[END_REF] around the equilibrium, by introducing a convenient weight. More precisely, let f be such that

w(t, x) = 1 + ε f (t, x) B m-1 (x) ∀ (t, x) ∈ R + × R d ,
for some small ε > 0. By substituting this expression in ( 18) and neglecting higher order terms in ε as ε → 0, we formally obtain a linear equation for f ,

f t = (1 -m) |x| γ B m-2 ∇ • |x| -β B ∇f , (28)
where the r.h.s. involves a positive, self-adjoint operator on L 2 (R d , B 2-m |x| -γ dx) associated with the closure of the quadratic form defined by

I[φ] := (1 -m) R d |∇φ| 2 B dx |x| β ∀ φ ∈ C ∞ 0 (R d ) .
The functional I[φ] is the linearized version of the Fisher information I, divided by (1m). By means of the same heuristics, we can linearize the free energy F as well to get, up to a factor 1/m,

F[φ] := 1 2 R d φ 2 B 2-m dx |x| γ .
If f is a solution of ( 28) then it is straightforward to infer that it satisfies

d dt F[f (t)] = -I[f (t)] , (29)
which by the way could also have been obtained by linearizing [START_REF] Grillo | On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density[END_REF]. In the case m > m * the conservation of relative mass becomes, after linearization,

R d f (t, x) B 2-m dx |x| γ = 0 ∀ t ≥ 0 .
Hence, as a consequence of ( 27) and ( 29), we formally get the following exponential decay for the linearized free energy:

F[f (t)] ≤ e -2 (1-m) Λ t F[f (0)] ∀ t ≥ 0 .
• Comparing linear and nonlinear quantities. Our aim here is to proceed in a similar way as in [2, Sections 5 and 6.2] so as to compare the free energy and Fisher information F and I with their linearized versions F and I, respectively. This will then allow us to give a rigorous justification of the above exponential decay and to use such an information to infer a precise exponential decay for the relative error.

Let us consider g = (w -1) B m-1 .

For t 0 ≥ 0 large enough, we deduce from Proposition 14 the existence of h ∈ (0, 1/4) such that w(t) -1 L ∞ (R d ) ≤ h for any t ≥ t 0 . The next result, whose proof we omit since it is identical to the one of [2, Lemma 3], shows the free energy compares with the linearized free energy.

Lemma 15. Assume that m ∈ (0, 1). If w is a solution of (18) corresponding to an initial datum w 0 satisfying assumptions (H1")-(H2"), then there exists t 0 ≥ 0 such that

m (1 + h) m-2 F[w(t)] ≤ F[w(t)] ≤ m (1 -h) m-2 F[w(t)] ∀ t ≥ t 0 .
For simplicity we shall assume that t 0 = 0 from now on. We now state the analogue of [START_REF] Bonforte | Special fast diffusion with slow asymptotics: entropy method and flow on a Riemann manifold[END_REF]Lemma 5.4]. The proof is again identical to the one performed in the case (β, γ) = (0, 0), so we skip it.

Lemma 16. Assume that m ∈ (0, 1). If w is a solution of (18) corresponding to an initial datum w 0 satisfying assumptions (H1")-(H2"), then

w(t) -1 2-m 1-m L 2-m 1-m ,γ (R d ) ≤ K F[w(t)] ∀ t ≥ 0 ,
where K is a positive constant depending only on m, C 1 , C 2 .

The next step is to get a bound of the L ∞ norm of the relative error in terms of the free energy.

Lemma 17. Assume that m ∈ (0, 1). If w is a solution of (18) corresponding to an initial datum w 0 satisfying assumptions (H1")-(H2"), then the following estimates hold:

w(t) -1 2-m 1-m L 2-m 1-m ,γ (R d ) ≤ κ 0 F[w(t)] ≤ κ 0 F[w 0 ] ∀ t ≥ 0 ( 30
)
and sup

τ ≥t w(τ ) -1 L ∞ (R d ) ≤ κ ∞ sup τ ≥t F[w(t)] θ ≤ κ ∞ F[w 0 ] θ ∀ t ≥ 1 , (31)
where Proof. Estimate ( 30) is a direct consequence of Lemmas 15-16 and of the fact that the free energy is nonincreasing by [START_REF] Grillo | On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density[END_REF].

θ :=    (1-m) (2+β-γ) (1-m) (2+β)+2+β-γ if γ ∈ (0, d) , 1-m 2-m if γ ≤ 0 ,
As for [START_REF] Reyes | Long time behavior for the inhomogeneous PME in a medium with slowly decaying density[END_REF], let us first consider the case γ ≥ 0. In this range we deduce from (H1") that

|w(t, x) -1| ≤ C B 1-m (x) ∀ (t, x) ∈ R + × R d
for a constant C > 0 depending on m, C 1 and C 2 , and, as a consequence,

|w(t, x) -1| γ 2+β-γ ≤ κ 1 |x| γ ∀ (t, x) ∈ R + × R d (32)
for some κ 1 depending on m, γ, β, C 1 , C 2 . By combining [START_REF] Reyes | The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions[END_REF] with [START_REF] Rosenau | Nonlinear diffusion in a finite mass medium[END_REF] we deduce that

w(t) -1 2-m 1-m + γ 2+β-γ L 2-m 1-m + γ 2+β-γ (R d ) = w(t) -1 1 θ L 1 θ (R d ) ≤ κ 1 κ 0 F[w(t)] ≤ κ 1 κ 0 F[w 0 ] (33
) for all t ≥ 0. Hence, [START_REF] Reyes | Long time behavior for the inhomogeneous PME in a medium with slowly decaying density[END_REF] follows with κ ∞ = (κ 1 κ 0 ) θ C 1/d ν,0,1/θ K by using [START_REF] Grillo | Radial fast diffusion on the hyperbolic space[END_REF] and generalised interpolation inequalities due to Gagliardo and Nirenberg (see, e.g., [2, Section 3] or [START_REF] Bonforte | Special fast diffusion with slow asymptotics: entropy method and flow on a Riemann manifold[END_REF]Appendix A.3]):

sup τ ≥t w(τ ) -1 L ∞ (R d ) ≤ C d θd+ν ν,0,1/θ sup τ ≥t |w(τ ) -1| θd θd+ν C ν (R d ) sup τ ≥t w(τ ) -1 ν θd+ν L 1 θ (R d ) ≤ C d θd+ν ν,0,1/θ K θd θd+ν sup τ ≥t w(τ ) -1 θd θd+ν L ∞ (R d ) sup τ ≥t w(τ ) -1 ν θd+ν L 1 θ (R d ) (34)
for all t ≥ 1, where C ν,0,1/θ is a positive constant depending only on d, ν, θ.

Let us now deal with the case γ < 0, where inequality ( 32) is no longer valid, so we have to proceed in a different way. To this end, first of all note that by Hölder's interpolation we obtain

w(t)-1 L p (Br) ≤ Br |w(t, x) -1| 2-m 1-m dx |x| γ 1-m 2-m Br |x| γ p(1-m) 2-m-p(1-m) dx 1 p -1-m 2-m and
w(t)-1 L q (B c r ) ≤ B c r |w(t, x) -1| 2-m 1-m dx |x| γ 1-m 2-m B c r |x| γ q(1-m) 2-m-q(1-m) dx 1 q -1-m 2-m
for all r > 0 and p, q ∈ (0, 2-m 1-m ). In particular, in view of Lemma 16, there exist p (sufficiently close to 0), q (sufficiently close to 2-m 1-m ) and a positive constant D depending on d, m, γ, C 1 , C 2 , r, such that

w(t) -1 L p (Br) ≤ D F[w(t)] 1-m 2-m and w(t) -1 L q (B c r ) ≤ D F[w(t)] 1-m 2-m . (35)
Let φ λ be the same family of cut-off functions as in the proof of Proposition 10. It is clear that

|φ 2 (w(t) -1)| C ν (R d ) ≤ c |w(t) -1| C ν (R d ) + w(t) -1 L ∞ (R d )
and

|(1 -φ 1 ) (w(t) -1)| C ν (R d ) ≤ c |w(t) -1| C ν (R d ) + w(t) -1 L ∞ (R d )
for some c > 0 depending only on ν and φ. Thanks to [START_REF] Grillo | Radial fast diffusion on the hyperbolic space[END_REF], by applying [START_REF] Vázquez | Fundamental solution and long time behavior of the porous medium equation in hyperbolic space[END_REF] to the functions φ 2 (w(t) -1) and (1φ 1 ) (w(t) -1), we obtain sup

τ ≥t w(τ ) -1 L ∞ (B2) ≤ C dp d+νp ν,0,p c d d+νp (K + 1) d d+νp sup τ ≥t w(τ ) -1 d d+νp L ∞ (R d ) sup τ ≥t w(τ ) -1 νp d+νp L p (B4) and sup τ ≥t w(τ ) -1 L ∞ (B c 2 ) ≤ C dq d+νq ν,0,q c d d+νq (K + 1) d d+νq sup τ ≥t w(τ ) -1 d d+νq L ∞ (R d ) sup τ ≥t w(τ ) -1 νq d+νq L q (B c 1 )
for all t ≥ 1. Hence, by exploiting [START_REF] Witelski | Self-similar asymptotics for linear and nonlinear diffusion equations[END_REF] with r = 4 and r = 1 in the right-hand sides and summing up the two estimates, we end up with sup

τ ≥t w(τ ) -1 L ∞ (R d ) ≤ C dp d+νp ν,0,p c d d+νp (K + 1) d d+νp D νp d+νp sup τ ≥t w(τ ) -1 d d+νp L ∞ (R d ) sup τ ≥t F[w(τ )] 1-m 2-m νp d+νp + C dq d+νq ν,0,q c d d+νq (K + 1) d d+νq D νq d+νq sup τ ≥t w(τ ) -1 d d+νq L ∞ (R d ) sup τ ≥t F[w(τ )] 1-m 2-m νq d+νq .
This completes the proof of [START_REF] Reyes | Long time behavior for the inhomogeneous PME in a medium with slowly decaying density[END_REF]

with θ = 1-m 2-m .
Now we compare the Fisher information with its linearized version in the spirit of [2, Lemma 7] and [5, Lemma 5.1].

Lemma 18. Assume that m ∈ (0, 1). If w is a solution of (18) corresponding to an initial datum w 0 satisfying assumptions (H1")-(H2"), then

I[w(t)] ≤ (1 + h) 3-2m (1 -m) (1 -h) I[w(t)] + µ h h F[w(t)] for a.e. t > 0 , (36)
where µ h is such that

2 (1 -h) µ h = (2 + β -γ) 2 (2 -m) 2 (1 -m) (1 + h) 4-2m (1 -4h) 1 2 + 2 (3-m) 3 (1-h) 4-m h 2 .
Proof. The proof is similar to the one of [5, Lemma 5.1]: here we give some details for the reader's convenience. For the sake of greater readability we shall again omit time dependence.

To begin with, let us rewrite the Fisher information I as

I[w] := (1 -m) 2 R d w ∇ A(w) (w -1) B m-1 2 B dx |x| β ,
where we have set

A(w) := w m-1 -1 (m -1) (w -1) =: a(w) w -1 .
It is easy to check that lim w→1 A(w) = 1, A(w) > 0 and A(w) → 0 as w → ∞. Moreover,

A (w) = w m-2 -A(w) w -1 ≤ 0 ,
since the function a(w) is concave in w, so that its incremental quotient A(w) (evaluated at w = 1) is a nonincreasing function of w. In particular,

(1 + h) m-2 ≤ A(w) ≤ (1 -h) m-2 .
Similarly, it is straightforward to show that A (w) is bounded. Now let us set g = (w -1)B m-1 . Since (w -1)A (w) + A(w) = w m-2 , we get:

∇ A(w) (w -1) B m-1 = A(w) ∇g + A (w) g B 1-m ∇g + A (w) g 2 ∇ B 1-m = [A(w) + A (w) (w -1)] ∇g + A (w) g 2 ∇ B 1-m = w m-2 ∇g + A (w) g 2 ∇ B 1-m .
Using Young's inequality a b ≤ h a 2 + b 2 /4h (for all a, b ∈ R) and the bounds 1h ≤ w ≤ 1 + h, we get:

I[w] (1 -m) 2 = R d w w m-2 ∇g + A (w) g 2 ∇ B 1-m 2 B dx |x| β ≥ (1 -h) R d |∇g| 2 w 2m-3 B dx |x| β - 1 -4 h 4 h R d g 4 |A (w)| 2 w ∇ B 1-m 2 B dx |x| β ≥ 1 -h (1 + h) 3-2m R d |∇g| 2 B dx |x| β - (1 + h) (1 -4 h) 4 h R d g 4 |A (w)| 2 ∇ B 1-m 2 B dx |x| β
(in the last passage we have used the fact that h < 1/4). We have therefore established the inequality

I[g] ≤ (1 + h) 3-2m (1 -m) (1 -h) I[w] + (1 -m) (1 + h) 4-2m (1 -4 h) 4 h (1 -h) R d g 4 |A (w)| 2 ∇ B 1-m 2 B dx |x| β .
To complete the proof, it is enough to establish the inequality

R d g 4 |A (w)| 2 ∇ B 1-m 2 B dx |x| β ≤ Q R d g 2 B 2-m dx |x| γ with Q := (2 + β -γ) 2 (2 -m) 2 1 2 + 2 (3-m) 3 (1-h) 4-m h 2 h 2 .
To this end, we observe that

∇ B 1-m 2 B |x| β = (2 + β -γ) 2 |x| 2+β-2γ (C + |x| 2+β-γ ) 4 B ≤ (2 + β -γ) 2 |x| γ B 4-3m , so that R d g 4 |A (w)| 2 ∇ B 1-m 2 B dx |x| β ≤ (2 + β -γ) 2 R d g 4 |A (w)| 2 B 4-3m dx |x| γ .
By definition of g = (w -1) B m-1 , using Taylor expansions and the bounds on w, through elementary computations we deduce that

g 2 |A (w)| 2 ≤ B 2m-2 (2 -m) 2 1 2 + 2 (3 -m) 3 (1 -h) 4-m h 2 h 2 ,
which concludes the proof.

• Convergence with sharp rates. By means of the results of Section 3.2 we shall first obtain a global (namely involving F and I) inequality of Hardy-Poincaré type and then use it to get sharp rates of convergence for F[w(t)], which in turn will yield rates for the relative error in view of Lemma 17.

Lemma 19. Assume that m ∈ (0, 1), m = m * . If w is a solution of (18) corresponding to an initial datum w 0 satisfying assumptions (H1")-(H2"), then there holds

[2 (1 -m) Λ -ρ h h] F[w(t)] ≤ m 1 -m I[w(t)] for a.e. t > 0 , ( 37
)
where Λ is the best constant appearing in the Hardy-Poincaré inequality [START_REF] Kamin | Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density[END_REF],

ρ h := 4 (1 -m) (3 -m) Λ + (1 -h) 3-m (1 + h) 3-2m µ h
and µ h is the same quantity as in Lemma 18.

Proof. With no loss of generality we can assume that F[w(t)] = 0 (and so also F[w(t)] = 0 thanks to Lemma 15), otherwise there is nothing to prove. The Hardy-Poincaré inequality [START_REF] Kamin | Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density[END_REF] plus Lemmas 15 and 18 then yield

2 (1 -m) Λ ≤ I[w(t)] F[w(t)] ≤ m (1 + h) 3-2m (1 -m) (1 -h) 3-m I[w(t)] F[w(t)] + µ h h , which reads (1 -h) 3-m (1 + h) 3-2m [2 (1 -m) Λ -µ h h] ≤ m 1 -m I[w(t)] F[w(t)] .
Finally, since

d dh (1 -h) 3-m (1 + h) 3-2m = - (1 -h) 2-m (1 + h) 4-2m (6 -3 m + m h) ≥ -2 (3 -m) , so that (1 -h) 3-m (1 + h) 3-2m ≥ 1 -2 (3 -m) h , we can deduce that (1 -h) 3-m (1 + h) 3-2m [2 (1 -m) Λ -µ h h] ≥ 2 (1 -m) Λ -4 (1 -m) (3 -m) Λ + (1 -h) 3-m (1 + h) 3-2m µ h h .
This concludes the proof.

Proof of Theorem 4. Let m = m * and assume that w is a solution of (18) corresponding to an initial datum w 0 satisfying assumptions (H1")-(H2"). We have to prove that, for some constants K 0 , t 0 > 0 that depend on d, m, γ, β, C 1 , C, C 2 and w 0 , the decay estimate

F[w(t)] ≤ K 0 e -2 (1-m) Λ (t-t0) ∀ t ≥ t 0 (38)
holds. We split the proof in two steps: in the first one we provide a non-sharp exponential decay for F[w(t)], in the second one we use the latter to get the sharp rate. We adopt implicitly the same notations as in Lemma 19.

• Step 1. By Proposition 14 we know that h(t) := w(t) -1 ∞ → 0 as t → ∞.

According to Lemma 15, there exists t 0 > 0 such that h(t) ≤ 1/4 for any t ≥ t 0 , and we can additionally require that

inf t≥t0 2 (1 -m) Λ -ρ h(t) h(t) ≥ (1 -m) Λ .
By combining this information, ( 24) and (37), we obtain

d dt F[w(t)] = - m 1 -m I[w(t)] ≤ -(1 -m) Λ F[w(t)] for a.e. t > t 0 ,
which yields the exponential-decay estimate

F[w(t)] ≤ F[w(t 0 )] e -(1-m) Λ (t-t0) ≤ F[w 0 ] e -(1-m) Λ (t-t0) ∀ t ≥ t 0 .
• Step 2. As a consequence of Lemma 17 and in particular [START_REF] Reyes | Long time behavior for the inhomogeneous PME in a medium with slowly decaying density[END_REF], we can infer that sup

τ ≥t h(τ ) ≤ κ ∞ F[w 0 ] θ e -θ (1-m) Λ (t-t0) ∀ t ≥ t 0 .
Moreover, it is clear that

0 < ρ ∞ := sup t≥t0 ρ h(t) < ∞ .
hence, inequality (37), which also holds with h = h(t), implies

2 (1 -m) Λ -ρ ∞ κ ∞ F[w 0 ] θ e -θ (1-m) Λ (t-t0) F[w(t)] ≤ m 1 -m I[w(t)]
for a.e. t > t 0 , so that by using again [START_REF] Grillo | On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density[END_REF] we end up with the differential inequality

d dt F[w(t)] ≤ -2 (1 -m) Λ -ρ ∞ κ ∞ F[w 0 ] θ e -θ (1-m) Λ (t-t0) F[w(t)]
for a.e. t > t 0 . An explicit integration then gives

F[w(t)] ≤ F[w 0 ] e ρ∞ κ∞ F [w 0 ] θ θ (1-m) Λ [1-e -θ (1-m) Λ (t-t 0 ) ] e -2 (1-m) Λ (t-t0)
for all t ≥ t 0 , namely (38) with

K 0 := F[w 0 ] e ρ∞ κ∞ F [w 0 ] θ θ (1-m) Λ
.

Theorem 5 follows as a straightforward consequence of Theorem 4, Lemma 17 and standard interpolation.

Additional results and comments.

4.1. Best matching, refined estimates and L 1,γ -convergence. The relative entropy to the best matching Barenblatt function is defined as

G[v] := inf µ>0 1 m -1 R d v m -B m µ -m B m-1 µ (v -B µ ) dx |x| γ ,
where the optimization is taken with respect to the scaling parameter µ > 0, that is, with respect to the set of the scaled Barenblatt functions

B µ (x) := µ d-γ B(µ x) ∀ x ∈ R d .
We start by a computation of the asymptotic rates which follows the line of thought developed in [START_REF] Dolbeault | Fast diffusion equations: matching large time asymptotics by relative entropy methods[END_REF][START_REF] Dolbeault | Best matching Barenblatt profiles are delayed[END_REF]. Also see [START_REF] Witelski | Self-similar asymptotics for linear and nonlinear diffusion equations[END_REF] for earlier considerations in this direction. An elementary calculation shows that in fact

G[v] = 1 m -1 R d v m -B m µ dx |x| γ , ( 39
)
where µ is the unique scaling parameter for which

R d |x| 2+β-γ v dx |x| γ = R d |x| 2+β-γ B µ dx |x| γ = µ -(2+β-γ) R d |x| 2+β-γ B dx |x| γ .
(40) This approach can be applied to any function v ∈ L 1,γ (R d ) and in particular to a t-dependent solution to [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF]. Moreover, we observe that

d dt R d |x| 2+β-γ v(t, x) dx |x| γ = -(2 + β -γ) (1 -m) 2 m G[v(t)] ≤ 0 .
Hence µ = µ (t) is monotone, with a positive limit as t → ∞, and this limit has to be equal to 1. Another remark is that

d dt G[v(t)] = - m 1 -m J [v(t)] ,
where J [v] denotes the relative Fisher information with respect the best matching Barenblatt function, defined as

J [v] := R d v ∇v m-1 -∇B m-1 µ 2 dx |x| β .
We can consider the linearized regime:

if v = B µ (1 + ε B 1-m µ f
), by neglecting higher order terms in ε, the moment condition (40) becomes

R d |x| 2+β-γ B 2-m µ f dx |x| γ = 0 . (41)
Let us recall the parameter ρ defined for the self-similar solution of the introduction by

1 ρ = (d -γ) (m -m c ) with m c = d-2-β d-γ
. With a simple scaling, we can also note that the spectral gap inequality of Proposition 3 is changed into

R d |∇f | 2 B µ dx |x| β ≥ Λ µ 1 ρ R d |f | 2 B 2-m µ dx |x| γ for any f ∈ L 2 (R d , B 2-m µ |x| -γ dx) such that R d f B 2-m µ |x| -γ dx = 0 if m > m * and (
41) holds. However, compared to Proposition 3, we obtain that the inequality holds with Λ = Λ ess if δ ≤ (n + 2)/2 and with Λ = Λ 0,1 if δ ≥ n/(2η), but with an improved spectral gap Λ > Λ 1,0 if (n + 2)/2 < δ < n/(2η), because of the orthogonality condition (41). See [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF]Appendix B] for details. Hence, by arguing as for the proof of Theorem 4, we obtain for the relative entropy G the following improved convergence rate.

G[v(t)] ≤ C e -2 (1-m) min{Λess,Λ0,1} t ∀ t ≥ 0 .
Next, we adapt the Csiszár-Kullback-Pinsker inequality of [START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF] to our setting. We recall that m

1 := d-γ d+2+β-2 γ . Lemma 21. Let d ≥ 1, m ∈ ( m 1 , 1) and assume that (6) holds. If v is a non- negative function in L 1,γ (R d ) such that G[v] is finte. If v L 1,γ (R d ) = M , then G[v] ≥ m 8 B m µ m L 1,γ (R d ) C(M ) v -B µ L 1,γ (R d ) + R d |x| 2+β-γ |v -B µ | dx |x| γ 2 .
The proof goes exactly along the lines of the one of [START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF]Theorem 4], except that the expression of B µ and the weight |x| -γ have to be taken into account. Details are left to the reader. Proposition 20 and Lemma 21 can be combined to give the result of convergence in L 1,γ (R d ) stated in Theorem 6. 4.2. Optimality of the constant on the curve of Felli and Schneider. For completeness, let us give the key idea of the proof of Theorem 1, (i), since the framework of the functional G is well adapted. In [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF], the proof is purely variational, but the flow setting is particularly convenient as we shall see next.

(0) = 1-m m (2 + β -γ) 2 such that J [v] ≥ Φ G[v] .
Proof. As in [4, Proposition 7], we notice that

J [v] -1-m m (2 + β -γ) 2 G[v] = 2 α p (m-1) 2 (2 m-1) 2 a D α w 2 L 2,d-n (R d ) + b w p+1 L p+1,d-n (R d ) -w 2 p n+2-p(n-2) n-p(n-4) L 2p,d-n (R d )
for some explicit constants a and b and for w = v m-1 2 . For a given function w ∈ C ∞ 0 (R d ), let us consider w µ (x) := µ n 2p w(µ x) for any x ∈ R d . An optimization with respect to µ as in [START_REF] Dolbeault | Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities[END_REF] shows the existence of a convex function Ψ such that

J [v] -1-m m (2 + β -γ) 2 G[v] ≥ Ψ 1 m-1 w p+1 L p+1,d-n (R d ) -B m-1 2 µ p+1 L p+1,d-n (R d )
.

The conclusion holds with Φ(s) = 1-m m (2 + βγ) 2 s + Ψ(s) using (39) and

w p+1 L p+1,d-n (R d ) = R d v m |x| -γ dx.
An elementary computation shows that Ψ is convex with Ψ(0) = Ψ (0) = 0.

Proof of Theorem 1, (i). Under the assumptions of Theorem 4, it is clear that the optimality in the inequality J 2) can be achieved only in the asymptotic regime, hence showing that Λ = 1 2 (2 + βγ) 2 /(1m) ≥ Λ 0,1 . On the other hand, if symmetry holds in [START_REF] Bonforte | Special fast diffusion with slow asymptotics: entropy method and flow on a Riemann manifold[END_REF], the opposite inequality also holds and hence we have equality. This characterizes the curve β = β F S (γ). This proof of course holds only for solutions corresponding to initial data such that (12) is satisfied, but an appropriate regularization allows us to conclude in the general case. 4.3. Concluding remarks. When (β, γ) = (0, 0), we know from [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF][START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF]

[v] ≥ 1-m m (2 + β -γ) 2 G[v] for a solution v = v(t) to (
that m 1 -m K(M ) = 2 (1 -m) Λ with Λ = Λ 0,1 ,
so that the global rate is the same as the asymptotic one obtained by linearization, and the corresponding eigenspace can be identified by considering the translations of the Barenblatt profiles. When (β, γ) = (0, 0), we may wonder when Λ = Λ 0,1 . Using the results of [4, Lemma 8], we can deduce that this holds whenever η = 1, which means α = α FS or, equivalently, β = β FS (γ).

As mentioned in the Introduction, in the case (β, γ) = (0, 0) Theorem 5 provides a better rate of convergence for the relative error with respect to the one obtained in [START_REF] Blanchet | Asymptotics of the fast diffusion equation via entropy estimates[END_REF]; in particular, we have the same rate for all L q norms with q ∈ 2-m 1-m , ∞ . However, in the case γ ∈ (0, d) the rate in [START_REF] Dolbeault | Weighted interpolation inequalities: a perturbation approach[END_REF] still depends on q. To some extent, this has to be expected. Indeed, as soon as γ > 0, it can easily be shown that Gagliardo-Nirenberg interpolation inequalities of the type of (34) fail if in the righthand side one puts an L p,γ norm. Since such inequalities are key in order to turn the decay of the free energy (38) into a uniform decay, the only way we can exploit them, as it is clear from the proof of Lemma 17, is by bounding a non-weighted norm of the relative error with a weighted norm or the free energy, like in [START_REF] Vázquez | The porous medium equation[END_REF] x |x| β v dx |x| γ = 0 . This moment corresponds to the eigenfunction f 0,1 , up to a multiplication by a constant, if β + 1 = α η. The reader is invited to check that this is possible if and only if β = 0. See [4, Appendix B] for technical details. When (β, γ) = (0, 0), the lowest moments are clearly associated with eigenspaces of the linearized evolution operator and responsible for the asymptotic rates of convergence of the evolution equation. If (β, γ) = (0, 0), the interpretation is not as straightforward.

Appendix. Hölder regularity at the origin for a degenerate/singular linear problem. First of all we observe that, to our purposes, it is convenient to change variables as in [4, Section 3.3], so that v(t, r, ω) = z(t, s, ω) with s = r α transforms (2) into z t -D * α z D α z m-1 -|x| 2 = 0 upon defining D * α as the adjoint to D α on L 2 (R d , |x| n-d dx), where the parameters α and n are as in [START_REF] Dolbeault | L q -functional inequalities and weighted porous media equations[END_REF] and

D α z := α ∂z ∂s , 1 s ∇ ω z .
In this regard, let us recall here some basic facts taken from [4, Section 3.3]. If f and g are respectively a vector-valued function and a scalar-valued function, then

R d f • (D α g) |x| n-d dx = R d (D * α f ) g |x| n-d dx .
In other words, if we take a representation of f adapted to spherical coordinates, that is s = |x| and ω = x/s, and consider

f s := f • ω and f ω := f -f s ω, then D * α f = -α s 1-n ∂ ∂s s n-1 f s - 1 s ∇ ω • f ω ,
where ∇ ω denotes the gradient with respect to angular derivatives only. In particular,

D * α [z 1 D α z 2 ] = -D α z 1 • D α z 2 + z 1 D * α (D α z 2 ) with -D * α (D α z 2 ) = α 2 s n-1 ∂ ∂s s n-1 ∂z 2 ∂s + 1 s 2 ∆ ω z 2
where ∆ ω represents the Laplace-Beltrami operator acting on ω ∈ S d-1 .

The advantage of resorting to this change of variables is that we can transform a problem with two different weights |x| -γ and |x| -β into a problem with two weights that are equal to |x| n-d . It is remarkable that Barenblatt-type stationary solutions [START_REF] Dolbeault | Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF] are transformed into the standard Barenblatt profiles

B = C + |x| 2 1 m-1 ∀ x ∈ R d .
Details on the change of variables can be found in [START_REF] Bonforte | Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities[END_REF]Section 2.3]. With regards to the purpose of this Appendix, the main interest of the change of variables is that it allows to use standard intrinsic cylinders. Given (t 0 , x 0 ) ∈ R + × R d and r > 0, let Q r (t 0 , x 0 ) := (t, x) ∈ R + × R d : t 0 -2 r 2 < t < t 0 , |xx 0 | < 2 r , Q + r (t 0 , x 0 ) := (t, x) ∈ R + × R d : t 0 -1 4 r 2 < t < t 0 , |xx 0 | < 1 2 r , Q - r (t 0 , x 0 ) := (t, x) ∈ R + × R d : t 0 -7 8 r 2 < t < t 0 -5 8 r 2 , |xx 0 | < 1 2 r . See Fig. 2. As a straightforward consequence of the above definitions, there holds Q r (t 0 , x 0 )

Q + r (t 0 , x 0 ) Q - r (t 0 , x 0 ) Figure 2.
The intrinsic cylinders Q r (t 0 , x 0 ), Q + r (t 0 , x 0 ) and Q - r (t 0 , x 0 ).

Q r/4 (t 0 , x 0 ) ⊂ Q + r (t 0 , x 0 ). The above cylinders are the same as the classical parabolic cylinders: having same weights gives the same scaling properties as in the non-weighted case, as first remarked in [START_REF] Chiarenza | A remark on a Harnack inequality for degenerate parabolic equations[END_REF].

Our aim here is to study the local Hölder regularity for solutions to a weighted linear problem of the form

u t + D * α a (D α u + B u) = 0 in R + × R d (42)
for some functions a and B which depend on (t, x) ∈ R + × R d . By following the ideas of F. Chiarenza and R. Serapioni in [START_REF] Chiarenza | A remark on a Harnack inequality for degenerate parabolic equations[END_REF], we start by establishing a parabolic Harnack inequality, through a weighted Moser iteration.

Proposition 23 (A parabolic Harnack inequality). Assume that a is locally bounded and bounded away from zero and that B is locally bounded in R + × R d . Let d ≥ 2, α > 0 and n > d. If u is a bounded positive solution of (42), then for all (t 0 , x 0 ) ∈ R + × R d and r > 0 such that Q r (t 0 , x 0 ) ⊂ R + × B 1 , we have sup

Q - r (t0,x0) u ≤ H inf Q + r (t0,x0)
u .

The constant H > 1 depends only on the local bounds on the coefficients a, B and on d, α, and n.

Proof. The proof follows the lines of [8, Theorem 2.1] with minor modifications. Let us emphasize the main adaptations. We observe that a critical Caffarelli-Kohn-Nirenberg inequality can be rewritten after the change of variables s = r α as

R d |w| 2 n n-2 |x| n-d dx n-2 n ≤ K n,α R d |D α w| 2 |x| n-d dx ∀ w ∈ C ∞ 0 (R d )
and is actually scale invariant. See [START_REF] Dolbeault | Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces[END_REF]Inequality 3.2] for details, including symmetry issues and the computation of K n,α in the symmetry range. This inequality plays the same role as the one of [START_REF] Chiarenza | A remark on a Harnack inequality for degenerate parabolic equations[END_REF]Lemma 1.1]. Then the proof follows upon replacing ∇ by D α . The term B u is in fact of lower order, since it is locally bounded: it can easily be reabsorbed into the energy estimates. By translating the intrinsic cylinders with respect to t by r 2 , we achieve the conclusion.

The Harnack inequality of Proposition 23 implies a Hölder continuity, by adapting the classical method à la De Giorgi to our weighted framework. Proof. We fix t 0 ≥ 1, r ∈ (0, 1/2) and denote for simplicity Q r := Q r (t 0 , 0) and Q ± r := Q ± r (t 0 , 0). Let us introduce the following quantities: We apply Proposition 23 to the nonnegative solution M 2ru to obtain

M 2r -m - r = sup Q - r (M 2r -u) ≤ H inf Q + r (M 2r -u) = H (M 2r -M + r ) .
Similarly, by using um 2r we obtain the inequality M - rm 2r ≤ H (m + rm 2r ) which, summed up with the previous inequality, gives

H (M + r -m + r ) + M - r -m - r ≤ (H -1) (M 2r -m 2r
) . Notice that we can always assume that H > 1. Using Q r/4 ⊂ Q + r , we conclude that osc

Q r/4 u ≤ osc Q + r u = M + r -m + r ≤ H -1 H (M 2r -m 2r ) = H -1 H osc Q2r u .
Without loss of generality we can assume that H/(H-1) ≤ 8: a well-known iteration technique (see, e.g., [START_REF] Giusti | Direct methods in the calculus of variations[END_REF]Lemma 6.1]) then shows that osc In particular we deduce that |u(t 0 )| C ν (B 1/2 ) ≤ K u L ∞ ((t0-1/2,t0)×B1) . Now note that, as a trivial consequence of Proposition 23 (just replace the inf with the sup in the r.h.s.), there holds u L ∞ ((t0-1/2,t0)×B1) ≤ H u L ∞ ((t+2,t+3)×B4) , which concludes the proof with K = K H.

Since the change of variables s = r α transforms Hölder functions into Hölder functions (but of course not C 1 ), as a direct consequence of Corollary 24 we have an analogous result for the original (linear) equation. where ν and K depend only on the local bounds on the coefficients a, B and on d, γ, β. Here α is given by (13).

(

  d-γ) (p-1) p [d+β+2-2 γ-p (d-β-2)] and parameters β, γ, p subject to d ≥ 2 , γ ∈ (-∞, d) , γ -2 < β < d-2 d γ(6)and p ∈ (1, p ] with p := d-γ d-β-2

3. 1 .

 1 Global regularity estimates: convergence without rates. a) From local to global estimates. By exploiting similar scaling techniques as in [2, Section 2.4], we use the regularity results of the Appendix in order to get global regularity estimates for w.

  and the positive constants κ 0 and κ ∞ depend on d, m, γ, β, C 1 , C, C 2 .

Proposition 20 .

 20 Let d ≥ 2 and assume that[START_REF] Bonforte | Global positivity estimates and Harnack inequalities for the fast diffusion equation[END_REF] holds, m ∈ (0, 1), m = m * . If m ∈ (0, m * ), we assume that (v 0 -B) ∈ L 1,γ (R d ), while we choose C = C(M ) if m > m * .With same notations as in Proposition 3, if v solves (2) and (12) holds, then there exists a positive constant C such that

Lemma 22 .

 22 Under the assumptions of Theorem 1, there exists a convex function Φ with Φ(0) = 0 and Φ

Corollary 24 (

 24 Hölder regularity at the origin I). Under the same assumptions as in Proposition 23, there exist ν ∈ (0, 1) and K > 0 such that|u(t)| C ν (B 1/2 ) ≤ K u L ∞ ((t+2,t+3)×B4) ∀ t ≥ 1 ,where ν and K depend only on the constant H > 1 of Proposition 23.

  Qr u ≤ C (2 r) ν osc Q 1/2 u ∀ r ∈ (0, 1/2] ,with ν := log(H/(H -1))/ log 8 ∈ (0, 1) and C > 0 depending only on H. A standard covering argument thus yields uniform Hölder continuity on smaller cylinders, namely|u| C ν,ν/2 (Qr) ≤ 2 ν K u L ∞ (Q 1/2 ) ∀ r ∈ (0, 1/4] ,where K > 0 is another constant that depends only on H and we set |u| C ν,ν/2 (Qr) := sup (t,x), (τ,y)∈Qr |u(t, x)u(τ, y)| (|x -y| 2 + |tτ |) ν/2 .

Corollary 25 (

 25 Hölder regularity at the origin II). Assume that d, β and γ comply with[START_REF] Bonforte | Global positivity estimates and Harnack inequalities for the fast diffusion equation[END_REF]. If u is a bounded positive solution of|x| -γ u t = ∇ • |x| -β a(t, x) (∇u + B(t, x) u) in R + × R d ,where a is locally bounded and bounded away from 0, B is locally bounded in R + ×R d . Then there exist ν ∈ (0, 1) and K > 0 such that|u(t)| C ν (B 2 -1/α ) ≤ K u L ∞ ((t+2,t+3)×B 2 2/α ) ∀ t ≥ 1 ,

Acknowledgments. This research has been partially supported by the projects STAB (J.D., B.N.) and Kibord (J.D.) of the French National Research Agency (ANR). M.B. has been funded by Project MTM2011-24696 and MTM2014-52240-P (Spain). This work has begun while M.B. and M.M. were visiting J.D. and B.N. in 2014. M.B. thanks the University of Paris 1 for inviting him. M.M. has been partially funded by the National Research Project "Calculus of Variations" (PRIN 2010-11, Italy) and by the "Università Italo-Francese / Université Franco-Italienne" (Bando Vinci 2013). J.D. also thanks the University of Pavia for support.

c 2016 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.


Rachid Dris

Johnny Gasperi

Vincent Rocher

Mohamed Saad

Bruno Tassin

Microplastics and urban water

Previous investigations in the Seine River

Less than 2 fragments/m 3 Up to 400 fibers/m 3

  particles/m2/day] Daily rainfall (mm) 11 Results -Atmospheric fallout • Microplastics in rainwater? Daily rainfall (mm) Atmospheric fallout (fibers/day/m 2 ) 12 Results -Rainwater Stormwater at the outlet of an urban catchment • The fibers either come from the runoff or are directly washed down by rain Between 28 and 60 fibers/