Main Results

Our main result is concerned with the sharp rate at which a solution v of the rescaled Eq. 5 converges to the generalized Barenblatt profile V_D given by formula 4 in the whole range m < 1. Convergence is measured in terms of the relative entropy given by the formula for all m ≠ 0 (modified as mentioned for m = 0). In order to get such convergence, we need the following assumptions on the initial datum v₀ associated to [5]:

(H1) V_D0 ≤ v₀ ≤ V_D1 for some D₀ > D₁ > 0,

(H2) if d≥3 and m ≤ m_∗, (v₀ - V_D) is integrable for a suitable D∈[D₁,D₀].

The case m = m_∗ will be discussed later. Besides, if m > m_∗, we define D as the unique value in [D₁,D₀] such that .

The precise meaning of what “sharp rate” means will be discussed at the end of this paper. As in ref. 4, we can deduce from Theorem 1 rates of convergence in more standard norms, namely, in L^q(dx) for q≥ max{1,2d(1 - m)/[2(2 - m) + d(1 - m)]}, or in C^k by interpolation. Moreover, by undoing the time-dependent change of variables [3], we can also deduce results on the intermediate asymptotics for the solution of Eq. 1; to be precise, we can get rates of decay of u(τ,y) - R(τ)^-dU_D,T(τ,y) as τ → +∞ if m∈[m_c,1), or as τ → T if m∈(-∞,m_c).

It is worth spending some words on the basin of attraction of the Barenblatt solutions U_D,T given by [2]. Such profiles have two parameters: D corresponds to the mass while T has the meaning of the extinction time of the solution for m < m_c and of a time-delay parameter otherwise. Fix T and D, and consider first the case m_∗ < m < 1. The basin of attraction of U_D,T contains all solutions corresponding to data which are trapped between two Barenblatt profiles U_D0,T(0,·),U_D1,T(0,·) for the same value of T and such that for some D∈[D₁,D₀]. If m < m_∗, the basin of attraction of a Barenblatt solution contains all solutions corresponding to data which, besides being trapped between U_D0,T and U_D1,T, are integrable perturbations of U_D,T(0,·).

Now, let us give an idea of the proof of Theorem 1. First assume that D = 1 (this entails no loss of generality). On , we shall therefore consider the measure dμ_α≔h_αdx, where the weight h_α is the Barenblatt profile, defined by h_α(x)≔(1 + |x|²)^α, with α = 1/(m - 1) < 0, and study on the weighted space L²(dμ_α) the operator which is such that , e.g., on . This operator appears in the linearization of [5] if, at a formal level, we expand in terms of ε, small, and only keep the first-order terms: The convergence result of Theorem 1 follows from the energy analysis of this equation based on the Hardy–Poincaré inequalities that are described below. Let us fix some notations. For d≥3, let us define α_∗≔-(d - 2)/2 corresponding to m = m_∗; two other exponents will appear in the analysis, namely, m₁≔(d - 1)/d with corresponding α₁ = -d, and m₂≔d/(d + 2) with corresponding α₂ = -(d + 2)/2. We have m_∗ < m_c < m₂ < m₁ < 1. Similar definitions for d = 2 give m_∗ = -∞, so that α_∗ = 0, as well as m_c = 0, and m₁ = m₂ = 1/2. For the convenience of the reader, Table 1 summarizes the key values of the parameter m and the corresponding values of α.

The Hardy–Poincaré inequalities [7] share many properties with Hardy’s inequalities, because of homogeneity reasons. A simple scaling argument indeed shows that holds for any f∈H¹((D + |x|²)^αdx) and any D≥0, under the additional conditions and D > 0 if α < α_∗. In other words, the optimal constant, Λ_α,d, does not depend on D > 0 and the assumption D = 1 can be dropped without consequences. In the limit D → 0, they yield weighted Hardy-type inequalities (cf. refs. 6 and 7).

Theorem 2 has been proved in (4) for m < m_∗. The main improvement of this report compared to refs. 4 and 8 is that we are able to give the value of the sharp constants also in the range (m_∗,1). These constants are deduced from the spectrum of the operator , that we shall study below (see Fig. 1).

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Fig. 1.

Spectrum of as a function of α, for d = 5.

It is relatively easy to obtain the classical decay rates of the linear case in the limit m → 1 by a careful rescaling such that weights become proportional to powers of the modified expression (1 + (1 - m)|x|²)^-1/(1-m). In the limit case, we obtain the Poincaré inequality for the Gaussian weight. As for the evolution equation, the time also has to be rescaled by a factor (1 - m). We leave the details to the reader. See ref. 9 for further considerations on associated functional inequalities.

Brief Historical Overview

The search for sharp decay rates in fast diffusion equations has been extremely active over the last three decades. Once plain convergence of the suitably rescaled flow toward an asymptotic profile is established (cf. refs. 1 and 2 for m > m_c and refs. 4 and 10 for m ≤ m_c), getting the rates is the next step in the asymptotic analysis. An important progress was achieved by Del Pino and Dolbeault (11) by identifying sharp rates of decay for the relative entropy, that had been introduced earlier by Newman (12) and Ralston (13). The analysis in ref. 11 uses the optimal constants in Gagliardo–Nirenberg inequalities, and these constants are computed. Carrillo and Toscani (14) gave a proof of decay based on the entropy/entropy-production method of Bakry and Emery, and established an analogue of the Csiszár–Kullback inequality which allows to control the convergence in L¹(dx), in case m > 1. Otto (15) then made the link with gradient flows with respect to the Wasserstein distance, and Cordero-Erausquin et al. (16) gave a proof of Gagliardo–Nirenberg inequalities using mass transportation techniques.

The condition m≥(d - 1)/d≕m₁ was definitely a strong limitation to these first approaches, except maybe for the entropy/entropy-production method. Gagliardo–Nirenberg inequalities degenerate into a critical Sobolev inequality for m = m₁, whereas the displacement convexity condition requires m≥m₁. It was a puzzling question to understand what was going on in the range m_c < m < m₁, and this has been the subject of many contributions. Because one is interested in understanding the convergence toward Barenblatt profiles, a key issue is the integrability of these profiles and their moments, in terms of m. To work with Wasserstein’s distance, it is crucial to have second moments bounded, which amounts to request m > d/(d + 2)≕m₂ for the Barenblatt profiles. The contribution of Denzler and McCann (17, 18) enters in this context. Another, weaker, limitation appears when one only requires the integrability of the Barenblatt profiles, namely m > m_c. Notice that the range [m_c,1) is also the range for which L¹(dx) initial data give rise to solutions which preserve the mass and globally exist; see, for instance, refs. 1 and 5.

It was therefore natural to investigate the range m∈(m_c,1) with entropy estimates. This has been done first by linearizing around the Barenblatt profiles in refs. 19 and 20, and then a full proof for the nonlinear flow was done by Carrillo and Vázquez (21). A detailed account for these contributions and their motivations can be found in the survey paper, ref. 22. Compared to classical approaches based on comparison, as in the book in ref. 5, a major advantage of entropy techniques is that they combine very well with L¹(dx) estimates if m > m_c, or relative mass estimates otherwise; see ref. 4.

The picture for m ≤ m_c turns out to be entirely different and more complicated, and it was not considered until quite recently. First of all, many classes of solutions vanish in finite time, which is a striking property that forces us to change the concept of asymptotic behavior from large-time behavior to behavior near the extinction time. On the other hand, L¹(dx) solutions lose mass as time evolves. Moreover, the natural extensions of Barenblatt’s profiles make sense but these profiles have two interesting properties: They vanish in finite time and they do not have finite mass.

There is a large variety of possible behaviors and many results have been achieved, such as the ones described in ref. 5 for data which decay strongly as |x| → ∞. However, as long as one is interested in solutions converging toward Barenblatt profiles in self-similar variables, there were some recent results on plain convergence: a paper of Daskalopoulos and Sesum (10), using comparison techniques, and in two contributions involving the authors of this paper, using relative entropy methods, see refs. 4 and 8. This last approach proceeds further into the description of the convergence by identifying a suitable weighted linearization of the relative entropy. In the appropriate space, L²(dμ_α-1), with the notations of Theorem 2, it gives rise to an exponential convergence after rescaling. This justifies the heuristic computation which relates Theorems 1 and 2, and allows to identify the sharp rates of convergence. The point of this paper is to explicitly state and prove such rates in the whole range m < 1.

Relative Entropy and Linearization

The strategy developed in ref. 4 is based on the extension of the relative entropy of Ralston and Newmann (12, 13), which can be written in terms of w = v/V_D as For simplicity, assume m ≠ 0. Notice that . Let be the generalized relative Fisher information. If v is a solution of [5], then [8]and, as a consequence, for all m < 1. The method is based on Theorem 2 and uniform estimates that relate linear and nonlinear quantities. Following refs. 4 and 23, we can first estimate from below and above the entropy in terms of its linearization, which appears in [7]: [9]where , , , and h≔ max{h₂,1 = h₁}. We notice that h(t) → 1 as t → +∞. Similarly, the generalized Fisher information satisfies the bounds [10]where and d(1 - m)[(h₂/h₁)^2(2-m) - 1] ≤ d(1 - m)[h^4(2-m) - 1]≕Y(h). Notice that X(1) = Y(1) = 0. Joining these inequalities with the Hardy–Poincaré inequality of Theorem 2 gives [11]as soon as 0 < h < h_∗≔ min{h > 0: Λ_α,d - Y(h)≥0}. On the other hand, uniform relative estimates hold, according to ref. 23 [formula (5.33)]: for some , [12]Summarizing, we end up with a system of nonlinear differential inequalities, with h as above and, at least for any t > t_∗, t_∗ > 0 large enough, [13]Gronwall-type estimates then show that This completes the proof of Theorem 1 for m ≠ 0. The adaptation to the logarithmic nonlinearity is left to the reader. Results in ref. 4 are improved in two ways: a time-dependent estimate of h is used in place of h(0), and the precise expression of the rate is established. One can actually get a slightly more precise estimate by coupling [12] and [13].

Operator Equivalence: Spectrum of

An important point of this paper is the computation of the spectrum of for any α < 0. This spectrum was only partially understood in refs. 4 and 8. In particular, the existence of a spectral gap was established for all α ≠ α_∗ = (2 - d)/2, but its value was not stated for all values of α.

Denzler and McCann (17, 18) formally linearized the fast diffusion flow (considered as a gradient flow of the entropy with respect to the Wasserstein distance) in the framework of mass transportation, in order to guess the asymptotic behavior of the solutions of [1]. This leads to a different functional setting, with a different linearized operator, . They performed the detailed analysis of its spectrum for all m∈(m_c,1), but the justification of the nonlinear asymptotics could not be completed due to the difficulties of the functional setting, especially in the very fast diffusion range.

Our approach is based on relative entropy estimates and the Hardy–Poincaré inequalities of Theorem 2. The asymptotics have readily been justified in ref. 4. The operator can be initially defined on . To construct a self-adjoint extension of such an operator, one can consider the quadratic form , where (·,·) denotes the scalar product on L²(dμ_α-1). Standard results show that such a quadratic form is closable, so that its closure defines a unique self-adjoint operator, its Friedrich’s extension, still denoted by the same symbol for brevity. The operator is different: It is obtained by taking the operator closure of , initially defined on , in the Hilbert space . Because of the Hardy–Poincaré inequality, defines a norm. Denote by 〈·,·〉 the corresponding scalar product and notice that for any .

The proof is based on the construction of a suitable unitary operator U: H^1,∗(dμ_α) → L²(dμ_α-1), such that . We claim that is the requested unitary operator. By definition, is a form core of , and, as a consequence, the identity has to be established only for functions . Because , we get where we have used the properties U^∗ = U^-1 and . This unitary equivalence between and implies the identity of their spectra.

We may now proceed with the presentation of the actual values of the spectrum by extending the results of ref. 18. According to ref. 24, the spectrum of the Laplace–Beltrami operator on S^d-1 is described by with ℓ = 0,1,2,… and with the convention M₀ = 1, and M₁ = 1 if d = 1. Using spherical coordinates and separation of variables, the discrete spectrum of is therefore made of the values of λ for which [14]has a solution on , in the domain of . The change of variables v(r) = r^ℓw(-r²) allows to express w in terms of the hypergeometric function with c = ℓ + d/2, a + b + 1 = ℓ + α + d/2, and ab = (2ℓα + λ)/4, as the solution for s = -r² of (see ref. 25). Based on refs. 8 and 18, we can state the following result.

Using Persson’s characterization of the continuous spectrum, see refs. 8 and 26, one can indeed prove that is the optimal constant in the following inequality: For any , The condition that the solution of [14] is in the domain of determines the eigenvalues. A more complete discussion of this topic can be found in ref. 18, which justifies the expression of the discrete spectrum.

Because α = 1/(m - 1), we may notice that for d≥2, α = -d (corresponding to -2α = λ₁₀ = λ₀₁ = -4α - 2d) and α = -(d + 2)/2 (corresponding to ), respectively, mean m = m₁ = (d - 1)/d and m = m₂ = d/(d + 2).

The above spectral results hold exactly in the same form when d = 2 (see ref. 18). Notice in particular that so that there is no equivalent of m_∗ for d = 2. With the notations of Theorem 2, α_∗ = 0. All results of Theorem 1 hold true under the sole assumption (H1).

In dimension d = 1, the spectral results are different (see ref. 18). The discrete spectrum is nonempty whenever α ≤ -1/2, that is m≥-1.

Critical Case

Because the spectral gap of tends to zero as m → m_∗, the previous strategy fails when m = m_∗ and one might expect a slower decay to equilibrium, sometimes referred as slow asymptotics. The following result has been proved in ref. 23.

Rates of convergence in L^q(dx), q∈(1,∞] follow. Notice that, in dimension d = 3 and 4, we have, respectively, m_∗ = -1 and m_∗ = 0. In the last case, Theorem 6 applies to the logarithmic diffusion.

The proof relies on identifying first the asymptotics of the linearized evolution. In this case, the bottom of the continuous spectrum of is zero. This difficulty is overcome by noticing that the operator on can be identified with the Laplace–Beltrami operator for a suitable conformally flat metric on , having positive Ricci curvature. Then the on-diagonal heat kernel of the linearized generator behaves like t^-d/2 for small t and like t^-1/2 for large t. The Hardy–Poincaré inequality is replaced by a weighted Nash inequality: There exists a positive continuous and monotone function on such that, for any nonnegative smooth function f with (recall that α_∗ - 1 = -d/2), The function behaves as follows: and . Only the first limit matters for the asymptotic behavior. Up to technicalities, inequality 13 is replaced by for some K > 0, t≥t₀ large enough, which allows to complete the proof.

Faster Convergence

A very natural issue is the question of improving the rates of convergence by imposing restrictions on the initial data. Results of this nature have been observed in ref. 19 in case of radially symmetric solutions, and are carefully commented in ref. 18. By locating the center of mass at zero, we are able to give an answer, which amounts to kill the λ₁₀ mode, whose eigenspace is generated by x↦x_i, i = 1,2,…,d. This is an improvement compared to the first result in this direction, which has been obtained by McCann and Slepčev (27), because we obtain an improved sharp rate of convergence of the solution of [5], as a consequence of the following improved Hardy–Poincaré inequality.

This covers the range m∈(m₁,1) with m₁ = (d - 1)/d.

Variational Approach to Sharpness

Recall that (d - 2)/d = m_c < m₁ = (d - 1)/d. The entropy / entropy-production inequality obtained in ref. 11 in the range m∈[m₁,1) can be written as and it is known to be sharp as a consequence of the optimality case in Gagliardo–Nirenberg inequalities. Moreover, equality is achieved if and only if v = V_D. The inequality has been extended into the range m∈(m_c,1) using the Bakry–Emery method, with the same constant 1/2, and again equality is achieved if and only if v = V_D, but sharpness of 1/2 is not as straightforward for m∈(m_c,m₁) as it is for m∈[m₁,1). The question of the optimality of the constant can be reformulated as a variational problem, namely to identify the value of the positive constant where the infimum is taken over the set of all functions such that and . Rephrasing the sharpness results, we know that if m∈(m₁,1) and if m∈(m_c,m₁). By taking and letting n → ∞, we get With the optimal choice for f, the above limit is less or equal than two. Because we already know that , this shows that for any m > m_c. It is quite enlightening to observe that optimality in the quotient gives rise to indetermination because both numerator and denominator are equal to zero when v = V_D. This also explains why it is the first-order correction which determines the value of , and, as a consequence, why the optimal constant is determined by the linearized problem.

When m ≤ m_c, the variational approach is less clear because the problem has to be constrained by a uniform estimate. Proving that any minimizing sequence is such that v_n/V_D - 1 converges, up to a rescaling factor, to a function f associated to the Hardy–Poincaré inequalities would be a significant step, except that one has to deal with compactness issues, test functions associated to the continuous spectrum, and a uniform constraint.

Sharp Rates of Convergence and Conjectures

In Theorem 1, we have obtained that the rate exp(-Λ_α,dt) is sharp. The precise meaning of this claim is that where the infimum is taken on the set of smooth, nonnegative bounded functions w such that ||w - 1||_L^∞(dx) ≤ h and such that is zero if d = 1, 2, and m < 1, or if d≥3 and m_∗ < m < 1, and it is finite if d≥3 and m < m_∗. Because, for a solution v(t,x) = w(t,x)V_D(x) of [5], [8] holds, by sharp rate we mean the best possible rate, which is uniform in t≥0. In other words, for any λ > Λ_α,d, one can find some initial datum in such that the estimate is wrong for some t > 0. We did not prove that the rate exp(-Λ_α,dt) is globally sharp in the sense that for some special initial data, decays exactly at this rate, or that , which is possibly less restrictive.

However, if m∈(m₁,1), m₁ = (d - 1)/d, then exp(-Λ_α,dt) is also a globally sharp rate, in the sense that the solution with initial datum u₀(x) = V_D(x + x₀) for any is such that decays exactly like exp(-Λ_α,dt). This formally answers the dilation-persistence conjecture as formulated in ref. 18. The question is still open when m ≤ m₁.

Another interesting issue is to understand if improved rates, that is, rates of the order of exp(-λ_ℓkt) with (ℓ,k) ≠ (0,0), (0, 1), (1, 0) are sharp or globally sharp under additional moment-like conditions on the initial data. It is also open to decide whether is sharp or globally sharp under the extra condition .