## ASYMPTOTICS OF THE FAST DIFFUSION EQUATION VIA ENTROPY ESTIMATES

Citations: | 15 - 5 self |

### BibTeX

@MISC{Blanchet_asymptoticsof,

author = {Adrien Blanchet and Matteo Bonforte and Jean Dolbeault and Gabriele Grillo and Juan and Luis V Ázquez},

title = {ASYMPTOTICS OF THE FAST DIFFUSION EQUATION VIA ENTROPY ESTIMATES},

year = {}

}

### OpenURL

### Abstract

Abstract. We consider non-negative solutions of the fast diffusion equation ut = ∆u m with m ∈ (0, 1), in the Euclidean space R d, d ≥ 3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to t → ∞ for m ≥ mc = (d−2)/d, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≥ mc, or close enough to the extinction time if m < mc. Such results are new in the range m ≤ mc where previous approaches fail. In the range mc < m < 1 we improve on known results. 1.

### Citations

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Citation Context ...e [25]. In such a case, E[v(t)|VD∗] ≤ E[v0|VD∗] e − 2 t ∀ t ≥ 0 . The limit case m = m1 corresponds to the critical Sobolev inequality whose optimal form was established by T. Aubin and G. Talenti in =-=[1, 44]-=-, while in the limit m → 1 one recovers Gross’ logarithmic Sobolev inequality, see [29, 25]. For m ∈ [m1, 1), F. Otto in [40] noticed that (1.4) can be interpreted as the gradient flow of the free ene... |

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Citation Context ...at the Barenblatt solution UD,T is integrable in y for m > mc, while the pseudo-Barenblatt solution corresponding to m ≤ mc is not integrable. Since much is known in the case m > mc, see for instance =-=[16, 25]-=- and [10, 11, 13, 14, 15, 27, 36, 45] for more complete results, the main novelty of our paper is concerned with the lower range m ≤ mc, which has several interesting new features. For instance, in th... |

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Citation Context ...> 0, that are obtained from the linearization of the relative entropy. In the limit D → 0, they yield the case corresponding to the weighted L2 norm of the Caffarelli-Kohn-Nirenberg inequalities, cf. =-=[12, 17]-=-. A final section, Appendix B, explains how to extend the results of this paper to the fast diffusion with exponents m ≤ 0. Note that the equation needs to be properly modified. The conclusion is that... |

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Citation Context ...ee [31, 32, 33] for results which seem closely related to ours, and [26] in the case m = (d − 2)/(d + 2). In particular we do not use the Bakry-Emery method introduced in [2], on which the results of =-=[15, 13, 36, 14, 16]-=- are based. We prove a conservation of relative mass, which allows us to remove the limitation m > mc. Neither mass transportation techniques nor Wasserstein distance are needed, although the approach... |

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Citation Context ...at the Barenblatt solution UD,T is integrable in y for m > mc, while the pseudo-Barenblatt solution corresponding to m ≤ mc is not integrable. Since much is known in the case m > mc, see for instance =-=[16, 25]-=- and [10, 11, 13, 14, 15, 27, 36, 45] for more complete results, the main novelty of our paper is concerned with the lower range m ≤ mc, which has several interesting new features. For instance, in th... |

35 |
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Citation Context ...ee [31, 32, 33] for results which seem closely related to ours, and [26] in the case m = (d − 2)/(d + 2). In particular we do not use the Bakry-Emery method introduced in [2], on which the results of =-=[15, 13, 36, 14, 16]-=- are based. We prove a conservation of relative mass, which allows us to remove the limitation m > mc. Neither mass transportation techniques nor Wasserstein distance are needed, although the approach... |

29 |
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Citation Context ... relative mass, which allows us to remove the limitation m > mc. Neither mass transportation techniques nor Wasserstein distance are needed, although the approach of Section A.3 is not unrelated, see =-=[8, 3, 4, 38]-=-. The paper is organized as follows. In Section 2, we extend the property of mass conservation, which holds only for m > mc, to a property of conservation of relative mass, see Proposition 2.3. This s... |

28 |
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Citation Context ...latt solution UD,T is integrable in y for m > mc, while the pseudo-Barenblatt solution corresponding to m ≤ mc is not integrable. Since much is known in the case m > mc, see for instance [16, 25] and =-=[10, 11, 13, 14, 15, 27, 36, 45]-=- for more complete results, the main novelty of our paper is concerned with the lower range m ≤ mc, which has several interesting new features. For instance, in the analysis in high space dimensions, ... |

20 |
Sobolev inequalities, the Poisson semigroup, and analysis on the sphere
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Citation Context ...̂ 1 ≤ d − 1 Sd−1 |∇ϑ u| 2 dϑ ̂ d−1 ∀ u ∈ H1(S ) . Here û := ∫ Sd−1 u ̂ dϑ. In the inequality, 1/(d − 1) is the optimal constant, as can be checked using spherical harmonic functions. See for instance =-=[5, 9, 43]-=-. The inequality itself can be recovered byASYMPTOTICS OF THE FAST DIFFUSION EQUATION 27 various methods. For example, using the inverse stereographic projection, see [37], the optimal Sobolev inequa... |

20 | Explicit constants for Rellich inequalities
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- 1998
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Citation Context ... ∫ 2 m |x|2) −1/(1−m) . Incidentally we observe that dµ = V 1−m D dν. To a R d g(x) dµ. Recall that m∗ = (d − 4)/(d − 2).s24 A. BLANCHET, M. BONFORTE, J. DOLBEAULT, G. GRILLO, AND J.L. V ÁZQUEZ As in =-=[24]-=-, we remark that ∆VD has a constant sign and get the estimate ∫ ∣ |g| 2 ∆VD dx ∣ = ∫ |g| 2 ∫ |∆VD| dx ≤ 4 R d Weights can be estimated on both sides of the inequality: |∆VD| V 2−m D |∇VD| 2 |∆VD| VD R... |

19 | The asymptotic behavior of gas in an n-dimensional porous medium - Friedman, Kamin - 1980 |

15 | Poincaré inequalities for linearizations of very fast diffusion equations
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Citation Context ...latt solution UD,T is integrable in y for m > mc, while the pseudo-Barenblatt solution corresponding to m ≤ mc is not integrable. Since much is known in the case m > mc, see for instance [16, 25] and =-=[10, 11, 13, 14, 15, 27, 36, 45]-=- for more complete results, the main novelty of our paper is concerned with the lower range m ≤ mc, which has several interesting new features. For instance, in the analysis in high space dimensions, ... |

14 |
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Citation Context ...r similar hypotheses (see [22, Theorem 1.4]). Actually they only prove the L ∞ convergence in case (ii) and the L1 ∩ L ∞ convergence in case (i). Our proof was obtained independently and announced in =-=[7]-=-. It is based on entropy estimates and paves the way to the sharper results on convergence with rates, which are the main purpose of the present paper. Assertion (iii) says that the convergence of (i)... |

14 |
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Citation Context ... = ∆ log u in dimension d ≥ 2, see e.g. [20, 21, 42, 48], which is the natural limiting equation to study in the limit m → 0. Also see [31, 32, 33] for results which seem closely related to ours, and =-=[26]-=- in the case m = (d − 2)/(d + 2). In particular we do not use the Bakry-Emery method introduced in [2], on which the results of [15, 13, 36, 14, 16] are based. We prove a conservation of relative mass... |

14 |
The Cauchy problem for ut = ∆u m when 0
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Citation Context ...> 0. We consider non-negative initial data and solutions. Existence and uniqueness of weak solutions of this problem with initial data in L1loc(R d ) was first proved by M.A. Herrero and M. Pierre in =-=[30]-=-. In the whole space, the behavior of the solutions is quite different in the parameter ranges mc < m < 1 and 0 < m < mc, the critical exponent being defined as mc := d − 2 d Note that mc > 0 only if ... |

14 |
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(Show Context)
Citation Context |

13 |
The maximal solution of the logarithmic fast diffusion equation in two space dimensions
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(Show Context)
Citation Context ...ise, which are still to be studied. In this paper, we leave apart several interesting questions, like the precise study of the case of m = m∗ or the equation ut = ∆ log u in dimension d ≥ 2, see e.g. =-=[20, 21, 42, 48]-=-, which is the natural limiting equation to study in the limit m → 0. Also see [31, 32, 33] for results which seem closely related to ours, and [26] in the case m = (d − 2)/(d + 2). In particular we d... |

11 |
Precise estimates on the rate at which certain diffusions tend to equilibrium, preprint
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Citation Context ...̂ 1 ≤ d − 1 Sd−1 |∇ϑ u| 2 dϑ ̂ d−1 ∀ u ∈ H1(S ) . Here û := ∫ Sd−1 u ̂ dϑ. In the inequality, 1/(d − 1) is the optimal constant, as can be checked using spherical harmonic functions. See for instance =-=[5, 9, 43]-=-. The inequality itself can be recovered byASYMPTOTICS OF THE FAST DIFFUSION EQUATION 27 various methods. For example, using the inverse stereographic projection, see [37], the optimal Sobolev inequa... |

10 | On the extinction profile of solutions to fast-diffusion
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- 2006
(Show Context)
Citation Context ...D∗ is determined by Assumption (H2). In this case, the presence of a perturbation f of VD∗ with nonzero mass, does not affect the asymptotic behavior of the solution at first order. In a recent paper =-=[22]-=-, P. Daskalopoulos and N. Sesum prove some of the results of Theorem 1.1 under similar hypotheses (see [22, Theorem 1.4]). Actually they only prove the L ∞ convergence in case (ii) and the L1 ∩ L ∞ co... |

10 |
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Citation Context ...and have various asymptotic behaviors depending on the initial data. Solutions with bounded and integrable initial data are described by self-similar solutions with so-called anomalous exponents, see =-=[34, 41]-=- and [47, Chapter 7]. Even for solutions with initial data not so far from a pseudo-Barenblatt solution, the asymptotic behavior may significantly differ from the behavior of a pseudo-Barenblatt solut... |

8 |
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Citation Context ...ζ), B(ζ)} 1 − m with [∫ r ∫ ζ A(ζ) := sup µ(s) ds r<ζ 0 r ] [∫ r ds , B(ζ) := sup ν(s) r>ζ ζ ds ν(s) ∫ +∞ r ] µ(s) ds . 2 m By convention, we take K(0) = 1−m B(0). The following result is inspired by =-=[3, 8, 19, 38]-=-. Proposition A.3. Let d ≥ 1. For any m ̸= m∗, If m ∈ (m∗, 1), then Cm,d ≥ Cm,1 ≤ K(0) and Cm,d ≤ max where, for any m ∈ (m∗, (d − 2)/(d − 1)), K(η) ≤ 8 m (1 − m) [d − 4 − m (d − 2)] 2 . { 2 K(η) , m ... |

7 | Modified logarithmic Sobolev inequalities on R
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Citation Context ... relative mass, which allows us to remove the limitation m > mc. Neither mass transportation techniques nor Wasserstein distance are needed, although the approach of Section A.3 is not unrelated, see =-=[8, 3, 4, 38]-=-. The paper is organized as follows. In Section 2, we extend the property of mass conservation, which holds only for m > mc, to a property of conservation of relative mass, see Proposition 2.3. This s... |

7 |
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Citation Context ...ise, which are still to be studied. In this paper, we leave apart several interesting questions, like the precise study of the case of m = m∗ or the equation ut = ∆ log u in dimension d ≥ 2, see e.g. =-=[20, 21, 42, 48]-=-, which is the natural limiting equation to study in the limit m → 0. Also see [31, 32, 33] for results which seem closely related to ours, and [26] in the case m = (d − 2)/(d + 2). In particular we d... |

6 | Self–similar solutions of a fast diffusion equation that do not conserve mass, Diff
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Citation Context ...and have various asymptotic behaviors depending on the initial data. Solutions with bounded and integrable initial data are described by self-similar solutions with so-called anomalous exponents, see =-=[34, 41]-=- and [47, Chapter 7]. Even for solutions with initial data not so far from a pseudo-Barenblatt solution, the asymptotic behavior may significantly differ from the behavior of a pseudo-Barenblatt solut... |

5 |
On the Cauchy problem for ut = ∆ log u in higher dimensions
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- 1999
(Show Context)
Citation Context ...ise, which are still to be studied. In this paper, we leave apart several interesting questions, like the precise study of the case of m = m∗ or the equation ut = ∆ log u in dimension d ≥ 2, see e.g. =-=[20, 21, 42, 48]-=-, which is the natural limiting equation to study in the limit m → 0. Also see [31, 32, 33] for results which seem closely related to ours, and [26] in the case m = (d − 2)/(d + 2). In particular we d... |

4 | Fine asymptotics near extinction and elliptic Harnack inequalities for the fast diffusion equation
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Citation Context |

4 |
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4 |
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3 | Hölder Estimates of Solutions of Singular Parabolic Equations with Measurable Coefficients - Chen, DiBenedetto - 1992 |

3 |
Quand est-ce que les bornes de Hardy permettent de calculer une constante de Poincaré exacte sur la droite
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Citation Context ... relative mass, which allows us to remove the limitation m > mc. Neither mass transportation techniques nor Wasserstein distance are needed, although the approach of Section A.3 is not unrelated, see =-=[8, 3, 4, 38]-=-. The paper is organized as follows. In Section 2, we extend the property of mass conservation, which holds only for m > mc, to a property of conservation of relative mass, see Proposition 2.3. This s... |

2 |
Eine Verschärfung der Poincaré-Ungleichung, Časopis Pěst
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(Show Context)
Citation Context ...̂ 1 ≤ d − 1 Sd−1 |∇ϑ u| 2 dϑ ̂ d−1 ∀ u ∈ H1(S ) . Here û := ∫ Sd−1 u ̂ dϑ. In the inequality, 1/(d − 1) is the optimal constant, as can be checked using spherical harmonic functions. See for instance =-=[5, 9, 43]-=-. The inequality itself can be recovered byASYMPTOTICS OF THE FAST DIFFUSION EQUATION 27 various methods. For example, using the inverse stereographic projection, see [37], the optimal Sobolev inequa... |

2 |
positivity estimates and Harnack inequalities for the fast diffusion equation
- Global
(Show Context)
Citation Context |

2 |
Classification of radially symmetric self-similar solutions of ut = ∆ log u in higher dimensions
- Hsu
(Show Context)
Citation Context ..., like the precise study of the case of m = m∗ or the equation ut = ∆ log u in dimension d ≥ 2, see e.g. [20, 21, 42, 48], which is the natural limiting equation to study in the limit m → 0. Also see =-=[31, 32, 33]-=- for results which seem closely related to ours, and [26] in the case m = (d − 2)/(d + 2). In particular we do not use the Bakry-Emery method introduced in [2], on which the results of [15, 13, 36, 14... |

2 |
time behavior of solutions of a singular diffusion equation
- Large
(Show Context)
Citation Context ..., like the precise study of the case of m = m∗ or the equation ut = ∆ log u in dimension d ≥ 2, see e.g. [20, 21, 42, 48], which is the natural limiting equation to study in the limit m → 0. Also see =-=[31, 32, 33]-=- for results which seem closely related to ours, and [26] in the case m = (d − 2)/(d + 2). In particular we do not use the Bakry-Emery method introduced in [2], on which the results of [15, 13, 36, 14... |

2 |
Asymptotic behavior for the porous medium equation posed in the whole space
- Vázquez
- 2003
(Show Context)
Citation Context ...n the last two decades, special attention has been given to the study of large time asymptotics of these equations, starting with the pioneering work of A. Friedman and S. Kamin [28] and completed in =-=[45]-=-, when m is in the range (mc, ∞). In those studies the class of non-negative, Date: April 14, 2007. Key words and phrases. Fast diffusion equation; self-similar solutions; asymptotic behavior; free en... |

2 |
decay estimates for nonlinear diffusion equations
- Smoothing
- 2006
(Show Context)
Citation Context ... with p∗ = d (1 − m)/2: then there exists a time T > 0 such that lim u(τ, y) = 0 . τ↗T Many computations are however similar in both ranges, from an algebraic point of view. We refer to the monograph =-=[47]-=- for a detailed discussion of the existence theory and references to the subject. The extension to exponents m ≤ 0 is also treated, and it is natural but it will not be the focus of this paper. In the... |