Special asymptotics for a critical fast diffusion equation

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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, ().
[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.
[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.
[13]	M. Fila and M. Winkler, Optimal rates of convergence to the singular Barenblatt profile for the fast diffusion equation,, preprint., ().
[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.
[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.
[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.
[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.

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

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