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August  2014, 7(4): 725-735. doi: 10.3934/dcdss.2014.7.725

Special asymptotics for a critical fast diffusion equation

Marek Fila 1, and  Hannes Stuke 2,

1. 

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

Institut für Mathematik, Freie Universität Berlin, 14195 Berlin, Germany

Received  September 2013 Revised  November 2013 Published  February 2014

We find a continuum of extinction rates of solutions of the Cauchy problem for the fast diffusion equation $u_\tau=\nabla\cdot(u^{m-1}\,\nabla u)$ with $m=m_*:=(n-4)/(n-2)$, here $n>2$ is the space-dimension. The extinction rates depend explicitly on the spatial decay rates of initial data and contain a logarithmic term.
Keywords: Fast diffusion, grow-up., extinction in finite time, nonlinear Fokker-Planck equation.
Mathematics Subject Classification: Primary: 35K65; Secondary: 35B4.
Citation: Marek Fila, Hannes Stuke. Special asymptotics for a critical fast diffusion equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 725-735. doi: 10.3934/dcdss.2014.7.725
References:
[1]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Nat. Bureau of Standards, Washington, D. C., 1964.  Google Scholar

[2]

J. G. Berryman and C. J. Holland, Stability of the separable solution for fast diffusion, Arch. Rat. Mech. Anal., 74 (1980), 379-388. doi: 10.1007/BF00249681.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

P. Daskalopoulos and N. Sesum, On the extinction profile of solutions to fast diffusion, J. Reine Angew. Math., 622 (2008), 95-119. doi: 10.1515/CRELLE.2008.066.  Google Scholar

[8]

M. del Pino and M. Sáez, On the extinction profile for solutions of $u_t=\Delta u^{(N-2)/(N+2)}$, Indiana Univ. Math. J., 50 (2001), 611-628. doi: 10.1512/iumj.2001.50.1876.  Google Scholar

[9]

E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dyn. Diff. Eq., 12 (2000), 647-673. doi: 10.1023/A:1026467729263.  Google Scholar

[10]

M. Fila, J. R. King and M. Winkler, Rate of convergence to Barenblatt profiles for the fast diffusion equation with a critical exponent,, preprint, ().   Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

V. A. Galaktionov and L. A. Peletier, Asymptotic behaviour near finite-time extinction for the fast diffusion equation, Arch. Rat. Mech. Anal., 139 (1997), 83-98. doi: 10.1007/s002050050048.  Google Scholar

[15]

J. R. King, Self-similar behaviour for the equation of fast nonlinear diffusion, Phil. Trans. Roy. Soc. Lond. A, 343 (1993), 337-375. doi: 10.1098/rsta.1993.0052.  Google Scholar

[16]

M. A. Peletier and H. Zhang, Self-similar solutions of a fast diffusion equation that do not conserve mass, Diff. Int. Equations, 8 (1995), 2045-2064.  Google Scholar

[17]

J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Notes in Maths. and its Applications, 33, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Nat. Bureau of Standards, Washington, D. C., 1964.  Google Scholar

[2]

J. G. Berryman and C. J. Holland, Stability of the separable solution for fast diffusion, Arch. Rat. Mech. Anal., 74 (1980), 379-388. doi: 10.1007/BF00249681.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

P. Daskalopoulos and N. Sesum, On the extinction profile of solutions to fast diffusion, J. Reine Angew. Math., 622 (2008), 95-119. doi: 10.1515/CRELLE.2008.066.  Google Scholar

[8]

M. del Pino and M. Sáez, On the extinction profile for solutions of $u_t=\Delta u^{(N-2)/(N+2)}$, Indiana Univ. Math. J., 50 (2001), 611-628. doi: 10.1512/iumj.2001.50.1876.  Google Scholar

[9]

E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dyn. Diff. Eq., 12 (2000), 647-673. doi: 10.1023/A:1026467729263.  Google Scholar

[10]

M. Fila, J. R. King and M. Winkler, Rate of convergence to Barenblatt profiles for the fast diffusion equation with a critical exponent,, preprint, ().   Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

V. A. Galaktionov and L. A. Peletier, Asymptotic behaviour near finite-time extinction for the fast diffusion equation, Arch. Rat. Mech. Anal., 139 (1997), 83-98. doi: 10.1007/s002050050048.  Google Scholar

[15]

J. R. King, Self-similar behaviour for the equation of fast nonlinear diffusion, Phil. Trans. Roy. Soc. Lond. A, 343 (1993), 337-375. doi: 10.1098/rsta.1993.0052.  Google Scholar

[16]

M. A. Peletier and H. Zhang, Self-similar solutions of a fast diffusion equation that do not conserve mass, Diff. Int. Equations, 8 (1995), 2045-2064.  Google Scholar

[17]

J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Notes in Maths. and its Applications, 33, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001.  Google Scholar

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