A continuum of extinction rates for the fast diffusion equation

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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

Abstract
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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. 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. 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. 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. 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. 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. 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. 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. 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, (2006). Google Scholar
[10]	J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'', Oxford Mathematical Monographs, (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. 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. 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. 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. 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. 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. 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. 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. 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, (2006). Google Scholar
[10]	J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'', Oxford Mathematical Monographs, (2007). Google Scholar

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