American Institute of Mathematical Sciences

Advanced
January  2015, 14(1): 107-119. doi: 10.3934/cpaa.2015.14.107

Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation

Marek Fila 1, and  Michael Winkler 2,

1. 

Department of Applied Mathematics and Statistics, Comenius University, 84248 Bratislava
2. 

Institut für Mathematik, Universität Paderborn, 33098 Paderborn

Received  January 2014 Revised  February 2014 Published  September 2014

We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast di usion equation with a critical exponent. We improve a previous result on slow convergence to Barenblatt pro les.
Keywords: nonlinear Fokker-Planck equation, extinction in finite time, slow convergence to Barenblatt profiles., Fast diffusion.
Mathematics Subject Classification: Primary: 35K65; Secondary: 35B4.
Citation: Marek Fila, Michael Winkler. Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 107-119. doi: 10.3934/cpaa.2015.14.107
References:
[1]

D. G. Aronson and P. Bénilan, Régularité des solutions de l'équations des milieux poreux dans $R^n$,, \emph{C. R. Acad. Sci. Paris, 288 (1979), 103. Google Scholar

[2]

A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates,, \emph{Arch. Rat. Mech. Anal.}, 191 (2009), 347. doi: 10.1007/s00205-008-0155-z. Google Scholar

[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,, \emph{Proc. Nat. Acad. Sciences}, 107 (2010), 16459. doi: 10.1073/pnas.1003972107. Google Scholar

[4]

M. Bonforte, G. Grillo and J. L. Vázquez, Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold,, \emph{Arch. Rat. Mech. Anal.}, 196 (2010), 631. doi: 10.1007/s00205-009-0252-7. Google Scholar

[5]

M. Bonforte, G. Grillo and J. L. Vázquez, Behaviour near extinction for the Fast Diffusion Equation on bounded domains,, \emph{J. Math. Pures Appl.}, 97 (2012), 1. doi: 10.1016/j.matpur.2011.03.002. Google Scholar

[6]

M. Fila, J. R. King and M. Winkler, Rate of convergence to Barenblatt profiles for the fast diffusion equation with a critical exponent,, \emph{J. London Math. Soc.}, 90 (2014), 167. doi: 10.1112/jlms/jdu025. Google Scholar

[7]

M. Fila and H. Stuke, Special asymptotics for a critical fast diffusion equation,, \emph{Discr. Cont. Dyn. Systems - S}, 7 (2014), 725. doi: 10.3934/dcdss.2014.7.725. Google Scholar

[8]

M. Fila, J. L. Vázquez and M. Winkler, A continuum of extinction rates for the fast diffusion equation,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 1129. doi: 10.3934/cpaa.2011.10.1129. Google Scholar

[9]

M. Fila, J. L. Vázquez, M. Winkler and E. Yanagida, Rate of convergence to Barenblatt profiles for the fast diffusion equation,, \emph{Arch. Rat. Mech. Anal.}, 204 (2012), 599. doi: 10.1007/s00205-011-0486-z. Google Scholar

[10]

M. Fila and M. Winkler, Optimal rates of convergence to the singular Barenblatt profile for the fast diffusion equation,, preprint., (). Google Scholar

show all references

References:
[1]

D. G. Aronson and P. Bénilan, Régularité des solutions de l'équations des milieux poreux dans $R^n$,, \emph{C. R. Acad. Sci. Paris, 288 (1979), 103. Google Scholar

[2]

A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates,, \emph{Arch. Rat. Mech. Anal.}, 191 (2009), 347. doi: 10.1007/s00205-008-0155-z. Google Scholar

[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,, \emph{Proc. Nat. Acad. Sciences}, 107 (2010), 16459. doi: 10.1073/pnas.1003972107. Google Scholar

[4]

M. Bonforte, G. Grillo and J. L. Vázquez, Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold,, \emph{Arch. Rat. Mech. Anal.}, 196 (2010), 631. doi: 10.1007/s00205-009-0252-7. Google Scholar

[5]

M. Bonforte, G. Grillo and J. L. Vázquez, Behaviour near extinction for the Fast Diffusion Equation on bounded domains,, \emph{J. Math. Pures Appl.}, 97 (2012), 1. doi: 10.1016/j.matpur.2011.03.002. Google Scholar

[6]

M. Fila, J. R. King and M. Winkler, Rate of convergence to Barenblatt profiles for the fast diffusion equation with a critical exponent,, \emph{J. London Math. Soc.}, 90 (2014), 167. doi: 10.1112/jlms/jdu025. Google Scholar

[7]

M. Fila and H. Stuke, Special asymptotics for a critical fast diffusion equation,, \emph{Discr. Cont. Dyn. Systems - S}, 7 (2014), 725. doi: 10.3934/dcdss.2014.7.725. Google Scholar

[8]

M. Fila, J. L. Vázquez and M. Winkler, A continuum of extinction rates for the fast diffusion equation,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 1129. doi: 10.3934/cpaa.2011.10.1129. Google Scholar

[9]

M. Fila, J. L. Vázquez, M. Winkler and E. Yanagida, Rate of convergence to Barenblatt profiles for the fast diffusion equation,, \emph{Arch. Rat. Mech. Anal.}, 204 (2012), 599. doi: 10.1007/s00205-011-0486-z. Google Scholar

[10]

M. Fila and M. Winkler, Optimal rates of convergence to the singular Barenblatt profile for the fast diffusion equation,, preprint., (). Google Scholar

[1]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017
[2]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485
[3]

Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016
[4]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[5]

Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028
[6]

Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845
[7]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056
[8]

Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008
[9]

Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028
[10]

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028
[11]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250
[12]

Marek Fila, Juan-Luis Vázquez, Michael Winkler. A continuum of extinction rates for the fast diffusion equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1129-1147. doi: 10.3934/cpaa.2011.10.1129
[13]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215
[14]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165
[15]

Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65
[16]

Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079
[17]

Simon Plazotta. A BDF2-approach for the non-linear Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2893-2913. doi: 10.3934/dcds.2019120
[18]

Florian Schneider, Andreas Roth, Jochen Kall. First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions. Kinetic & Related Models, 2017, 10 (4) : 1127-1161. doi: 10.3934/krm.2017044
[19]

John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371
[20]

Yun-Gang Chen, Yoshikazu Giga, Koh Sato. On instant extinction for very fast diffusion equations. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 243-250. doi: 10.3934/dcds.1997.3.243

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences