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

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.
Citation: Marek Fila, Michael Winkler. Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation. Communications on Pure and 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 $R^n$, C. R. Acad. Sci. Paris, Sér. A-B, 288 (1979), 103-105.

[2]

A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Rat. Mech. Anal., 191 (2009), 347-385. 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, Proc. Nat. Acad. Sciences, 107 (2010), 16459-16464. doi: 10.1073/pnas.1003972107.

[4]

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

[5]

M. Bonforte, G. Grillo and J. L. Vázquez, Behaviour near extinction for the Fast Diffusion Equation on bounded domains, J. Math. Pures Appl., 97 (2012), 1-38. doi: 10.1016/j.matpur.2011.03.002.

[6]

M. Fila, J. R. King and M. Winkler, Rate of convergence to Barenblatt profiles for the fast diffusion equation with a critical exponent, J. London Math. Soc., 90 (2014), 167-183. doi: 10.1112/jlms/jdu025.

[7]

M. Fila and H. Stuke, Special asymptotics for a critical fast diffusion equation, Discr. Cont. Dyn. Systems - S, 7 (2014), 725-735. doi: 10.3934/dcdss.2014.7.725.

[8]

M. Fila, J. L. Vázquez and M. Winkler, A continuum of extinction rates for the fast diffusion equation, Comm. Pure Appl. Anal., 10 (2011), 1129-1147. doi: 10.3934/cpaa.2011.10.1129.

[9]

M. Fila, J. L. Vázquez, M. Winkler and E. Yanagida, Rate of convergence to Barenblatt profiles for the fast diffusion equation, Arch. Rat. Mech. Anal., 204 (2012), 599-625. doi: 10.1007/s00205-011-0486-z.

[10]

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

show all references

References:
[1]

D. G. Aronson and P. Bénilan, Régularité des solutions de l'équations des milieux poreux dans $R^n$, C. R. Acad. Sci. Paris, Sér. A-B, 288 (1979), 103-105.

[2]

A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Rat. Mech. Anal., 191 (2009), 347-385. 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, Proc. Nat. Acad. Sciences, 107 (2010), 16459-16464. doi: 10.1073/pnas.1003972107.

[4]

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

[5]

M. Bonforte, G. Grillo and J. L. Vázquez, Behaviour near extinction for the Fast Diffusion Equation on bounded domains, J. Math. Pures Appl., 97 (2012), 1-38. doi: 10.1016/j.matpur.2011.03.002.

[6]

M. Fila, J. R. King and M. Winkler, Rate of convergence to Barenblatt profiles for the fast diffusion equation with a critical exponent, J. London Math. Soc., 90 (2014), 167-183. doi: 10.1112/jlms/jdu025.

[7]

M. Fila and H. Stuke, Special asymptotics for a critical fast diffusion equation, Discr. Cont. Dyn. Systems - S, 7 (2014), 725-735. doi: 10.3934/dcdss.2014.7.725.

[8]

M. Fila, J. L. Vázquez and M. Winkler, A continuum of extinction rates for the fast diffusion equation, Comm. Pure Appl. Anal., 10 (2011), 1129-1147. doi: 10.3934/cpaa.2011.10.1129.

[9]

M. Fila, J. L. Vázquez, M. Winkler and E. Yanagida, Rate of convergence to Barenblatt profiles for the fast diffusion equation, Arch. Rat. Mech. Anal., 204 (2012), 599-625. doi: 10.1007/s00205-011-0486-z.

[10]

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

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