American Institute of Mathematical Sciences

Advanced
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

[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]

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
[3]

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
[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]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021, 8 (4) : 381-402. doi: 10.3934/jdg.2021013
[7]

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
[8]

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
[9]

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
[10]

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
[11]

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

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
[13]

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
[14]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215
[15]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic & Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009
[16]

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
[17]

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
[18]

Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079
[19]

Simon Plazotta. A BDF2-approach for the non-linear Fokker-Planck equation. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2893-2913. doi: 10.3934/dcds.2019120
[20]

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

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy