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July  2011, 10(4): 1129-1147. doi: 10.3934/cpaa.2011.10.1129

A continuum of extinction rates for the fast diffusion equation

Marek Fila 1, , Juan-Luis Vázquez 2, and  Michael Winkler 3,

1. 

Department of Applied Mathematics and Statistics, Comenius University, 842 48 Bratislava
2. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid
3. 

Fakultät für Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany

Received  September 2010 Revised  November 2011 Published  April 2011

We find a continuum of extinction rates for solutions $u(y,\tau)\ge 0$ of the fast diffusion equation $u_\tau=\Delta u^m$ in a subrange of exponents $m\in (0,1)$. The equation is posed in $R^n$ for times up to the extinction time $T>0$. The rates take the form $\|u(\cdot,\tau)\|_\infty$ ~ $(T-\tau)^\theta$ for a whole interval of $\theta>0$. These extinction rates depend explicitly on the spatial decay rates of initial data.
Keywords: grow-up., extinction in finite time, Fast diffusion, nonlinear Fokker-Planck equation.
Mathematics Subject Classification: Primary: 35K65; Secondary: 35B4.
Citation: 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
References:
[1]

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

[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.  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, Proc. Nat. Acad. Sciences, 107 (2010), 16459-16464. 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, Arch. Rat. Mech. Anal., 196 (2010), 631-680. doi: 10.1007/s00205-009-0252-7.  Google Scholar

[5]

J. Denzler and R. J. McCann, Fast diffusion to self-Similarity: Complete spectrum, long-time asymptotics, and numerology, Arch. Rat. Mech. Anal., 175 (2005), 301-342. doi: 10.1007/s00205-004-0336-3.  Google Scholar

[6]

M. Fila, J. King, M. Winkler and E. Yanagida, Optimal lower bound of the grow-up rate for a supercritical parabolic equation, J. Diff. Equations, 228 (2006), 339-356. doi: 10.1016/j.jde.2006.01.019.  Google Scholar

[7]

M. Fila and M. Winkler, Rate of convergence to a singular steady state of a supercritical parabolic equation, J. Evol. Equations, 8 (2008), 673-692. doi: 10.1007/s00028-008-0400-9.  Google Scholar

[8]

M. Fila, M. Winkler and E. Yanagida, Grow-up rate of solutions for a supercritical semilinear diffusion equation, J. Diff. Equations, 205 (2004), 365-389. doi: 10.1016/j.jde.2004.03.009.  Google Scholar

[9]

J. L. Vázquez, "Smoothing and Decay Estimates for Nonlinear Diffusion Equations,'' Oxford Lecture Notes in Maths. and its Applications, vol. 33, Oxford University Press, Oxford, 2006.  Google Scholar

[10]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'' Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007.  Google Scholar

show all references

References:
[1]

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

[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.  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, Proc. Nat. Acad. Sciences, 107 (2010), 16459-16464. 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, Arch. Rat. Mech. Anal., 196 (2010), 631-680. doi: 10.1007/s00205-009-0252-7.  Google Scholar

[5]

J. Denzler and R. J. McCann, Fast diffusion to self-Similarity: Complete spectrum, long-time asymptotics, and numerology, Arch. Rat. Mech. Anal., 175 (2005), 301-342. doi: 10.1007/s00205-004-0336-3.  Google Scholar

[6]

M. Fila, J. King, M. Winkler and E. Yanagida, Optimal lower bound of the grow-up rate for a supercritical parabolic equation, J. Diff. Equations, 228 (2006), 339-356. doi: 10.1016/j.jde.2006.01.019.  Google Scholar

[7]

M. Fila and M. Winkler, Rate of convergence to a singular steady state of a supercritical parabolic equation, J. Evol. Equations, 8 (2008), 673-692. doi: 10.1007/s00028-008-0400-9.  Google Scholar

[8]

M. Fila, M. Winkler and E. Yanagida, Grow-up rate of solutions for a supercritical semilinear diffusion equation, J. Diff. Equations, 205 (2004), 365-389. doi: 10.1016/j.jde.2004.03.009.  Google Scholar

[9]

J. L. Vázquez, "Smoothing and Decay Estimates for Nonlinear Diffusion Equations,'' Oxford Lecture Notes in Maths. and its Applications, vol. 33, Oxford University Press, Oxford, 2006.  Google Scholar

[10]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'' Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007.  Google Scholar

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