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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

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