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March  2017, 10(1): 61-91. doi: 10.3934/krm.2017003

Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods

1. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain

2. 

Ceremade, UMR CNRS $n^{\circ }$ 7534, Université Paris-Dauphine, PSL Research University, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France

3. 

Dipartimento di Matematica Felice Casorati, Università degli Studi di Pavia, Via A. Ferrata 5,27100 Pavia, Italy

4. 

SAMM, Université Paris 1, 90, rue de Tolbiac, 75634 Paris Cedex 13, France

* Corresponding author

Received  February 2016 Revised  June 2016 Published  November 2016

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 purposPart Ⅱis 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.

Citation: Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods. Kinetic & Related Models, 2017, 10 (1) : 61-91. doi: 10.3934/krm.2017003
References:
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A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Hardy-Poincaréinequalities and applications to nonlinear diffusions, Comptes Rendus Mathématique, 344 (2007), 431–436, URL http://dx.doi.org/10.1016/j.crma.2007.01.011. doi: 10.1016/j.crma.2007.01.011.

[2]

A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Archive for Rational Mechanics and Analysis, 191 (2009), 347–385, URL http://dx.doi.org/10.1007/s00205-008-0155-z. doi: 10.1007/s00205-008-0155-z.

[3]

M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proceedings of the National Academy of Sciences, 107 (2010), 16459–16464, URL http://dx.doi.org/10.1073/pnas.1003972107. doi: 10.1073/pnas.1003972107.

[4]

M. Bonforte, J. Dolbeault, M. Muratori and B. Nazaret, Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in CaffarelliKohn-Nirenberg inequalities, to appear in Kinet. Relat. Models, Preprint hal-01279326 & arXiv: 1602.08319, (2016).

[5]

M. Bonforte, G. Grillo and J. L. Vázquez, Special fast diffusion with slow asymptotics: Entropy method and flow on a Riemann manifold, Arch. Ration. Mech. Anal., 196 (2010), 631–680, URL http://dx.doi.org/10.1007/s00205-009-0252-7. doi: 10.1007/s00205-009-0252-7.

[6]

M. Bonforte and J. L. Vázquez, Global positivity estimates and Harnack inequalities for the fast diffusion equation, J. Funct. Anal., 240 (2006), 399–428, URL http://dx.doi.org/10.1016/j.jfa.2006.07.009. doi: 10.1016/j.jfa.2006.07.009.

[7]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259–275, URL http://eudml.org/doc/89687.

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F. Chiarenza and R. Serapioni, A remark on a Harnack inequality for degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova, 73 (1985), 179–190, URL http://www.numdam.org/item?id=RSMUP_1985__73__179_0.

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M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9), 81 (2002), 847–875, URL http://dx.doi.org/10.1016/S0021-7824(02)01266-7. doi: 10.1016/S0021-7824(02)01266-7.

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J. Denzler, H. Koch and R. J. McCann, Higher-order time asymptotics of fast diffusion in Euclidean space: A dynamical systems approach, Mem. Amer. Math. Soc., 234 (2015), vi+81pp, URL https://dx.doi.org/10.1090/memo/1101 doi: 10.1090/memo/1101.

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J. Dolbeault, M. J. Esteban and M. Loss, Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces, to appear in Inventiones Mathematicae, URL http://link.springer.com/article/10.1007/s00222-016-0656-6, (2016)

[12]

J. Dolbeault, M. J. Esteban, M. Loss and M. Muratori, Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities, to appear in C. R. Mathématique, Preprint hal-01318727 & arXiv: 1605.06373, (2016). doi: 10.1016/j.crma.2017.01.004.

[13]

J. Dolbeault, I. Gentil, A. Guillin and F.-Y. Wang, Lq-functional inequalities and weighted porous media equations, Potential Anal., 28 (2008), 35–59, URL http://dx.doi.org/10.1007/s11118-007-9066-0. doi: 10.1007/s11118-007-9066-0.

[14]

J. Dolbeault, M. Muratori and B. Nazaret, Weighted interpolation inequalities: a perturbation approach, to appear in Mathematische Annalen, Preprint hal-01207009 & arXiv: 1509.09127, (2016). doi: 10.1007/s00208-016-1480-4.

[15]

J. Dolbeault and G. Toscani, Fast diffusion equations: matching large time asymptotics by relative entropy methods, Kinetic and Related Models, 4 (2011), 701–716, URL http://dx.doi.org/10.3934/krm.2011.4.701. doi: 10.3934/krm.2011.4.701.

[16]

J. Dolbeault and G. Toscani, Improved interpolation inequalities, relative entropy and fast diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 917–934, URL http://dx.doi.org/10.1016/j.anihpc.2012.12.004. doi: 10.1016/j.anihpc.2012.12.004.

[17]

J. Dolbeault and G. Toscani, Best matching Barenblatt profiles are delayed, Journal of Physics A: Mathematical and Theoretical, 48 (2015), 065206, 14pp, URL http://dx.doi.org/10.1088/1751-8113/48/6/065206 doi: 10.1088/1751-8113/48/6/065206.

[18]

J. Dolbeault and G. Toscani, Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays, Nonlinear Analysis: Theory, Methods & Applications, 138 (2016), 31–43, URL http://www.sciencedirect.com/science/article/pii/S0362546X15003880. doi: 10.1016/j.na.2015.11.012.

[19]

J. Dolbeault and G. Toscani, Stability results for logarithmic Sobolev and Gagliardo– Nirenberg inequalities, International Mathematics Research Notices, 2016 (2016), 473–498, URL http://imrn.oxfordjournals.org/content/early/2015/05/15/imrn.rnv131.abstract. doi: 10.1093/imrn/rnv131.

[20]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003, URL http://dx.doi.org/10.1142/9789812795557 doi: 10.1142/9789812795557.

[21]

G. Grillo and M. Muratori, Radial fast diffusion on the hyperbolic space, Proc. Lond. Math. Soc. (3), 109 (2014), 283–317, URL http://dx.doi.org/10.1112/plms/pdt071. doi: 10.1112/plms/pdt071.

[22]

G. Grillo, M. Muratori and M. M. Porzio, Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities, Discrete Contin. Dyn. Syst., 33 (2013), 3599–3640, URL http://dx.doi.org/10.3934/dcds.2013.33.3599. doi: 10.3934/dcds.2013.33.3599.

[23]

G. Grillo, M. Muratori and F. Punzo, On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density, Discrete Contin. Dyn. Syst., 35 (2015), 5927–5962, URL http://dx.doi.org/10.3934/dcds.2015.35.5927. doi: 10.3934/dcds.2015.35.5927.

[24]

M. A. Herrero and M. Pierre, The Cauchy problem for $u_t=Δ u^m$ when $0< m < 1$, Trans. Amer. Math. Soc., 291 (1985), 145–158. doi: 10.1090/S0002-9947-1985-0797051-0.

[25]

R. G. Iagar and A. Sánchez, Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density, Nonlinear Anal., 102 (2014), 226–241, URL http://dx.doi.org/10.1016/j.na.2014.02.016. doi: 10.1016/j.na.2014.02.016.

[26]

S. Kamin, G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density, Discrete Contin. Dyn. Syst., 26 (2010), 521–549, URL http://dx.doi.org/10.3934/dcds.2010.26.521. doi: 10.3934/dcds.2010.26.521.

[27]

S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math., 34 (1981), 831–852, URL http://dx.doi.org/10.1002/cpa.3160340605. doi: 10.1002/cpa.3160340605.

[28]

S. Nieto and G. Reyes, Asymptotic behavior of the solutions of the inhomogeneous porous medium equation with critical vanishing density, Commun. Pure Appl. Anal., 12 (2013), 1123–1139, URL http://dx.doi.org/10.3934/cpaa.2013.12.1123. doi: 10.3934/cpaa.2013.12.1123.

[29]

G. Reyes and J. L. Vázquez, The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions, Commun. Pure Appl. Anal., 7 (2008), 1275-1294. doi: 10.3934/cpaa.2008.7.1275.

[30]

G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal., 8 (2009), 493–508, URL http://dx.doi.org/10.3934/cpaa.2009.8.493. doi: 10.3934/cpaa.2009.8.493.

[31]

P. Rosenau and S. Kamin, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113–127, URL http://dx.doi.org/10.1002/cpa.3160350106. doi: 10.1002/cpa.3160350106.

[32]

J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007, Mathematical theory.

[33]

J. L. Vázquez, Fundamental solution and long time behavior of the porous medium equation in hyperbolic space, J. Math. Pures Appl. (9), 104 (2015), 454–484, URL http://dx.doi.org/10.1016/j.matpur.2015.03.005. doi: 10.1016/j.matpur.2015.03.005.

[34]

T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math., 100 (1998), 153–193, URL http://dx.doi.org/10.1111/1467-9590.00074. doi: 10.1111/1467-9590.00074.

show all references

References:
[1]

A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Hardy-Poincaréinequalities and applications to nonlinear diffusions, Comptes Rendus Mathématique, 344 (2007), 431–436, URL http://dx.doi.org/10.1016/j.crma.2007.01.011. doi: 10.1016/j.crma.2007.01.011.

[2]

A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Archive for Rational Mechanics and Analysis, 191 (2009), 347–385, URL http://dx.doi.org/10.1007/s00205-008-0155-z. doi: 10.1007/s00205-008-0155-z.

[3]

M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proceedings of the National Academy of Sciences, 107 (2010), 16459–16464, URL http://dx.doi.org/10.1073/pnas.1003972107. doi: 10.1073/pnas.1003972107.

[4]

M. Bonforte, J. Dolbeault, M. Muratori and B. Nazaret, Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in CaffarelliKohn-Nirenberg inequalities, to appear in Kinet. Relat. Models, Preprint hal-01279326 & arXiv: 1602.08319, (2016).

[5]

M. Bonforte, G. Grillo and J. L. Vázquez, Special fast diffusion with slow asymptotics: Entropy method and flow on a Riemann manifold, Arch. Ration. Mech. Anal., 196 (2010), 631–680, URL http://dx.doi.org/10.1007/s00205-009-0252-7. doi: 10.1007/s00205-009-0252-7.

[6]

M. Bonforte and J. L. Vázquez, Global positivity estimates and Harnack inequalities for the fast diffusion equation, J. Funct. Anal., 240 (2006), 399–428, URL http://dx.doi.org/10.1016/j.jfa.2006.07.009. doi: 10.1016/j.jfa.2006.07.009.

[7]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259–275, URL http://eudml.org/doc/89687.

[8]

F. Chiarenza and R. Serapioni, A remark on a Harnack inequality for degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova, 73 (1985), 179–190, URL http://www.numdam.org/item?id=RSMUP_1985__73__179_0.

[9]

M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9), 81 (2002), 847–875, URL http://dx.doi.org/10.1016/S0021-7824(02)01266-7. doi: 10.1016/S0021-7824(02)01266-7.

[10]

J. Denzler, H. Koch and R. J. McCann, Higher-order time asymptotics of fast diffusion in Euclidean space: A dynamical systems approach, Mem. Amer. Math. Soc., 234 (2015), vi+81pp, URL https://dx.doi.org/10.1090/memo/1101 doi: 10.1090/memo/1101.

[11]

J. Dolbeault, M. J. Esteban and M. Loss, Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces, to appear in Inventiones Mathematicae, URL http://link.springer.com/article/10.1007/s00222-016-0656-6, (2016)

[12]

J. Dolbeault, M. J. Esteban, M. Loss and M. Muratori, Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities, to appear in C. R. Mathématique, Preprint hal-01318727 & arXiv: 1605.06373, (2016). doi: 10.1016/j.crma.2017.01.004.

[13]

J. Dolbeault, I. Gentil, A. Guillin and F.-Y. Wang, Lq-functional inequalities and weighted porous media equations, Potential Anal., 28 (2008), 35–59, URL http://dx.doi.org/10.1007/s11118-007-9066-0. doi: 10.1007/s11118-007-9066-0.

[14]

J. Dolbeault, M. Muratori and B. Nazaret, Weighted interpolation inequalities: a perturbation approach, to appear in Mathematische Annalen, Preprint hal-01207009 & arXiv: 1509.09127, (2016). doi: 10.1007/s00208-016-1480-4.

[15]

J. Dolbeault and G. Toscani, Fast diffusion equations: matching large time asymptotics by relative entropy methods, Kinetic and Related Models, 4 (2011), 701–716, URL http://dx.doi.org/10.3934/krm.2011.4.701. doi: 10.3934/krm.2011.4.701.

[16]

J. Dolbeault and G. Toscani, Improved interpolation inequalities, relative entropy and fast diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 917–934, URL http://dx.doi.org/10.1016/j.anihpc.2012.12.004. doi: 10.1016/j.anihpc.2012.12.004.

[17]

J. Dolbeault and G. Toscani, Best matching Barenblatt profiles are delayed, Journal of Physics A: Mathematical and Theoretical, 48 (2015), 065206, 14pp, URL http://dx.doi.org/10.1088/1751-8113/48/6/065206 doi: 10.1088/1751-8113/48/6/065206.

[18]

J. Dolbeault and G. Toscani, Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays, Nonlinear Analysis: Theory, Methods & Applications, 138 (2016), 31–43, URL http://www.sciencedirect.com/science/article/pii/S0362546X15003880. doi: 10.1016/j.na.2015.11.012.

[19]

J. Dolbeault and G. Toscani, Stability results for logarithmic Sobolev and Gagliardo– Nirenberg inequalities, International Mathematics Research Notices, 2016 (2016), 473–498, URL http://imrn.oxfordjournals.org/content/early/2015/05/15/imrn.rnv131.abstract. doi: 10.1093/imrn/rnv131.

[20]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003, URL http://dx.doi.org/10.1142/9789812795557 doi: 10.1142/9789812795557.

[21]

G. Grillo and M. Muratori, Radial fast diffusion on the hyperbolic space, Proc. Lond. Math. Soc. (3), 109 (2014), 283–317, URL http://dx.doi.org/10.1112/plms/pdt071. doi: 10.1112/plms/pdt071.

[22]

G. Grillo, M. Muratori and M. M. Porzio, Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities, Discrete Contin. Dyn. Syst., 33 (2013), 3599–3640, URL http://dx.doi.org/10.3934/dcds.2013.33.3599. doi: 10.3934/dcds.2013.33.3599.

[23]

G. Grillo, M. Muratori and F. Punzo, On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density, Discrete Contin. Dyn. Syst., 35 (2015), 5927–5962, URL http://dx.doi.org/10.3934/dcds.2015.35.5927. doi: 10.3934/dcds.2015.35.5927.

[24]

M. A. Herrero and M. Pierre, The Cauchy problem for $u_t=Δ u^m$ when $0< m < 1$, Trans. Amer. Math. Soc., 291 (1985), 145–158. doi: 10.1090/S0002-9947-1985-0797051-0.

[25]

R. G. Iagar and A. Sánchez, Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density, Nonlinear Anal., 102 (2014), 226–241, URL http://dx.doi.org/10.1016/j.na.2014.02.016. doi: 10.1016/j.na.2014.02.016.

[26]

S. Kamin, G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density, Discrete Contin. Dyn. Syst., 26 (2010), 521–549, URL http://dx.doi.org/10.3934/dcds.2010.26.521. doi: 10.3934/dcds.2010.26.521.

[27]

S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math., 34 (1981), 831–852, URL http://dx.doi.org/10.1002/cpa.3160340605. doi: 10.1002/cpa.3160340605.

[28]

S. Nieto and G. Reyes, Asymptotic behavior of the solutions of the inhomogeneous porous medium equation with critical vanishing density, Commun. Pure Appl. Anal., 12 (2013), 1123–1139, URL http://dx.doi.org/10.3934/cpaa.2013.12.1123. doi: 10.3934/cpaa.2013.12.1123.

[29]

G. Reyes and J. L. Vázquez, The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions, Commun. Pure Appl. Anal., 7 (2008), 1275-1294. doi: 10.3934/cpaa.2008.7.1275.

[30]

G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal., 8 (2009), 493–508, URL http://dx.doi.org/10.3934/cpaa.2009.8.493. doi: 10.3934/cpaa.2009.8.493.

[31]

P. Rosenau and S. Kamin, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113–127, URL http://dx.doi.org/10.1002/cpa.3160350106. doi: 10.1002/cpa.3160350106.

[32]

J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007, Mathematical theory.

[33]

J. L. Vázquez, Fundamental solution and long time behavior of the porous medium equation in hyperbolic space, J. Math. Pures Appl. (9), 104 (2015), 454–484, URL http://dx.doi.org/10.1016/j.matpur.2015.03.005. doi: 10.1016/j.matpur.2015.03.005.

[34]

T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math., 100 (1998), 153–193, URL http://dx.doi.org/10.1111/1467-9590.00074. doi: 10.1111/1467-9590.00074.

Figure 1.  The spectrum of $\mathcal L$ as a function of $\delta=\frac1{1-m}$, with $n=5$. The essential spectrum corresponds to the grey area, and its bottom is determined by the parabola $\delta\mapsto\Lambda_{\rm ess}(\delta)$. The two eigenvalues $\Lambda_{0,1}$ and $\Lambda_{1,0}$ are given by the plain, half-lines, away from the essential spectrum. Note that solutions of the eigenvalue problem exist for any value of $\delta$ but may not be in the domain of the operator or below the essential spectrum and are then represented as dotted half-lines. See [4,Appendix B] for a discussion of the values of $\delta_1$, $\delta_2$, ... $\delta_5$. The right figure is an enlargement of the left one. This configuration is not generic: see [4,Fig. 5] for other cases.
Figure 2.  The intrinsic cylinders $Q_r(t_0,x_0)$, $Q^+_r(t_0,x_0)$ and $Q^-_r(t_0,x_0)$.
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