Asymptotic large time behavior of singular solutions of the fast diffusion equation

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November 2017, 37(11): 5943-5977. doi: 10.3934/dcds.2017258

Asymptotic large time behavior of singular solutions of the fast diffusion equation

Kin Ming Hui ^1, and Soojung Kim ^2,,

1.	Institute of Mathematics, Academia Sinica, Taipei, Taiwan
2.	Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China

* Corresponding author: Soojung Kim

Received December 2016 Revised June 2017 Published July 2017

We study the asymptotic large time behavior of singular solutions of the fast diffusion equation

$u_t=Δ u^m$

in

$({\mathbb R}^n\setminus\{0\})×(0, ∞)$

in the subcritical case

$0<m<\frac{n-2}{n}$

,

$n≥3$

. Firstly, we prove the existence of the singular solution

$u$

of the above equation that is trapped in between self-similar solutions of the form of

$t^{-α} f_i(t^{-β}x)$

,

$i=1, 2$

, with the initial value

$u_0$

satisfying

$A_1|x|^{-γ}≤ u_0≤ A_2|x|^{-γ}$

for some constants

$A_2>A_1>0$

and

$\frac{2}{1-m}<γ<\frac{n-2}{m}$

, where

$β:=\frac{1}{2-γ(1-m)}$, $α:=\frac{2\beta-1}{1-m}, $

and the self-similar profile

$f_i$

satisfies the elliptic equation

$Δ f^m+α f+β x· \nabla f=0 \,\,\,\,\,\,\mbox{ in ${\mathbb R}^n\setminus\{0\}$}$

with $\lim_{|x|\to0}|x|^{\frac{ α}{ β}}f_i(x)=A_i$ and $\lim_{|x|\to∞}|x|^{\frac{n-2}{m}}{f_i}(x)= D_{A_i} $ for some constants $D_{A_i}>0$. When $\frac{2}{1-m} < γ < n$, under an integrability condition on the initial value $u_0$ of the singular solution $u$, we prove that the rescaled function

$\tilde u(y, τ):= t^{\, α} u(t^{\, β} y, t),\,\,\,\,\,\, { τ:=\log t}, $

converges to some self-similar profile $f$ as $τ\to∞$.

Keywords: Existence, large time behavior, fast diffusion equation, singular solution, self-similar solution.

Mathematics Subject Classification: Primary: 35B35, 35B44, 35K55, 35K65.

Citation: Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

References:

[1]	D. G. Aronson, The porous medium equation, Nonlinear diffusion problems, (Montecatini Terme, 1985), 1-46, Lecture Notes in Math., 1224, Springer, Berlin, 1986. doi: 10.1007/BFb0072687.
[2]	A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal., 191 (2009), 347-385. doi: 10.1007/s00205-008-0155-z.
[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. Natl. Acad. Sci. USA, 107 (2010), 16459-16464. doi: 10.1073/pnas.1003972107.
[4]	E. Chasseigne and J. L. Vázquez, Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities, Arch. Ration. Mech. Anal., 164 (2002), 133-187. doi: 10.1007/s00205-002-0210-0.
[5]	P. Daskalopoulos and C. E. Kenig, Degenerate Diffusion: Initial Value Problems and Local Regularity Theory, EMS Tracts in Mathematics, 1. European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/033.
[6]	P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, arXiv: 1306.0859.
[7]	P. Daskalopoulos, M. del Pino and N. Sesum, Type Ⅱ ancient compact solutions to the Yamabe flow, J. Reine Angew. Math., (2015), http://dx.doi.org/10.1515/crelle-2015-0048 in press. doi: 10.1515/crelle-2015-0048.
[8]	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.
[9]	P. Daskalopoulos and N. Sesum, The classification of locally conformally flat Yamabe solitons, Adv. Math., 240 (2013), 346-369. doi: 10.1016/j.aim.2013.03.011.
[10]	M. Fila, J. L. Vázquez, M. Winkler and E. Yanagida, Rate of convergence to Barenblatt profiles for the fast diffusion equation, Arch. Ration. Mech. Anal., 204 (2012), 599-625. doi: 10.1007/s00205-011-0486-z.
[11]	M. Fila and M. Winkler, Optimal rates of convergence to the singular Barenblatt profile for the fast diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 309-324. doi: 10.1017/S0308210515000554.
[12]	M. Fila and M. Winkler, Rate of convergence to separable solutions of the fast diffusion equation, Israel J. Math., 213 (2016), 1-32. doi: 10.1007/s11856-016-1319-4.
[13]	M. Fila and M. Winkler, Slow growth of solutions of superfast diffusion equations with unbounded initial data, J. London Math. Soc.(2), 95 (2017), 659-683. doi: 10.1112/jlms.12029.
[14]	M. A. Herrero and M. Pierre, The Cauchy problem for $u_t = \Delta u^m$ when $0 < m < 1$, Trans. Amer. Math. Soc., 291 (1985), 145-158. doi: 10.1090/S0002-9947-1985-0797051-0.
[15]	S.Y. Hsu, Asymptotic profile of solutions of a singular diffusion equation as $t \to∞$, Nonlinear Anal., 48 (2002), 781-790. doi: 10.1016/S0362-546X(00)00214-5.
[16]	S. Y. Hsu, Singular limit and exact decay rate of a nonlinear elliptic equation, Nonlinear Anal., 75 (2012), 3443-3455. doi: 10.1016/j.na.2012.01.009.
[17]	S. Y. Hsu, Existence and asymptotic behaviour of solutions of the very fast diffusion equation, Manuscripta Math., 140 (2013), 441-460. doi: 10.1007/s00229-012-0576-8.
[18]	K. M. Hui, On some Dirichlet and Cauchy problems for a singular diffusion equation, Differential Integral Equations, 15 (2002), 769-804.
[19]	K. M. Hui, Singular limit of solutions of the very fast diffusion equation, Nonlinear Anal., 68 (2008), 1120-1147. doi: 10.1016/j.na.2006.12.009.
[20]	K. M. Hui, Asymptotic behaviour of solutions of the fast diffusion equation near its extinction time, J. Math. Anal. Appl., 454 (2017), 695-715. doi: 10.1016/j.jmaa.2017.05.006.
[21]	T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss. 132, Springer-Verlag, Berlin, New York, 1976.
[22]	O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Transl. Math. Mono. vol. 23, Amer. Math. Soc., Providence, R. I., U. S. A., 1968.
[23]	S. J. Osher and J. V. Ralston, L¹ stability of traveling waves with applications to convective porous media flow, Comm. Pure Appl. Math., 35 (1982), 737-749. doi: 10.1002/cpa.3160350602.
[24]	M. del Pino and M. Sáez, On the extinction profile for solutions of $u_t=\Delta u^{\frac{N-2}{N+2}}$, Indiana Univ. Math. J., 50 (2001), 611-628. doi: 10.1512/iumj.2001.50.1876.
[25]	J. L. Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl.(9), 71 (1992), 503-526.
[26]	J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Oxford Lecture Series in Mathematics and its Applications 33, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001.
[27]	J. L. Vázquez and M. Winkler, The evolution of singularities in fast diffusion equations: Infinite time blow-down, SIAM J. Math. Anal., 43 (2011), 1499-1535. doi: 10.1137/100809465.
[28]	R. Ye, Global existence and convergence of Yamabe flow, J. Differential Geom., 39 (1994), 35-50. doi: 10.4310/jdg/1214454674.

show all references

References:

[1]	D. G. Aronson, The porous medium equation, Nonlinear diffusion problems, (Montecatini Terme, 1985), 1-46, Lecture Notes in Math., 1224, Springer, Berlin, 1986. doi: 10.1007/BFb0072687.
[2]	A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal., 191 (2009), 347-385. doi: 10.1007/s00205-008-0155-z.
[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. Natl. Acad. Sci. USA, 107 (2010), 16459-16464. doi: 10.1073/pnas.1003972107.
[4]	E. Chasseigne and J. L. Vázquez, Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities, Arch. Ration. Mech. Anal., 164 (2002), 133-187. doi: 10.1007/s00205-002-0210-0.
[5]	P. Daskalopoulos and C. E. Kenig, Degenerate Diffusion: Initial Value Problems and Local Regularity Theory, EMS Tracts in Mathematics, 1. European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/033.
[6]	P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, arXiv: 1306.0859.
[7]	P. Daskalopoulos, M. del Pino and N. Sesum, Type Ⅱ ancient compact solutions to the Yamabe flow, J. Reine Angew. Math., (2015), http://dx.doi.org/10.1515/crelle-2015-0048 in press. doi: 10.1515/crelle-2015-0048.
[8]	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.
[9]	P. Daskalopoulos and N. Sesum, The classification of locally conformally flat Yamabe solitons, Adv. Math., 240 (2013), 346-369. doi: 10.1016/j.aim.2013.03.011.
[10]	M. Fila, J. L. Vázquez, M. Winkler and E. Yanagida, Rate of convergence to Barenblatt profiles for the fast diffusion equation, Arch. Ration. Mech. Anal., 204 (2012), 599-625. doi: 10.1007/s00205-011-0486-z.
[11]	M. Fila and M. Winkler, Optimal rates of convergence to the singular Barenblatt profile for the fast diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 309-324. doi: 10.1017/S0308210515000554.
[12]	M. Fila and M. Winkler, Rate of convergence to separable solutions of the fast diffusion equation, Israel J. Math., 213 (2016), 1-32. doi: 10.1007/s11856-016-1319-4.
[13]	M. Fila and M. Winkler, Slow growth of solutions of superfast diffusion equations with unbounded initial data, J. London Math. Soc.(2), 95 (2017), 659-683. doi: 10.1112/jlms.12029.
[14]	M. A. Herrero and M. Pierre, The Cauchy problem for $u_t = \Delta u^m$ when $0 < m < 1$, Trans. Amer. Math. Soc., 291 (1985), 145-158. doi: 10.1090/S0002-9947-1985-0797051-0.
[15]	S.Y. Hsu, Asymptotic profile of solutions of a singular diffusion equation as $t \to∞$, Nonlinear Anal., 48 (2002), 781-790. doi: 10.1016/S0362-546X(00)00214-5.
[16]	S. Y. Hsu, Singular limit and exact decay rate of a nonlinear elliptic equation, Nonlinear Anal., 75 (2012), 3443-3455. doi: 10.1016/j.na.2012.01.009.
[17]	S. Y. Hsu, Existence and asymptotic behaviour of solutions of the very fast diffusion equation, Manuscripta Math., 140 (2013), 441-460. doi: 10.1007/s00229-012-0576-8.
[18]	K. M. Hui, On some Dirichlet and Cauchy problems for a singular diffusion equation, Differential Integral Equations, 15 (2002), 769-804.
[19]	K. M. Hui, Singular limit of solutions of the very fast diffusion equation, Nonlinear Anal., 68 (2008), 1120-1147. doi: 10.1016/j.na.2006.12.009.
[20]	K. M. Hui, Asymptotic behaviour of solutions of the fast diffusion equation near its extinction time, J. Math. Anal. Appl., 454 (2017), 695-715. doi: 10.1016/j.jmaa.2017.05.006.
[21]	T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss. 132, Springer-Verlag, Berlin, New York, 1976.
[22]	O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Transl. Math. Mono. vol. 23, Amer. Math. Soc., Providence, R. I., U. S. A., 1968.
[23]	S. J. Osher and J. V. Ralston, L¹ stability of traveling waves with applications to convective porous media flow, Comm. Pure Appl. Math., 35 (1982), 737-749. doi: 10.1002/cpa.3160350602.
[24]	M. del Pino and M. Sáez, On the extinction profile for solutions of $u_t=\Delta u^{\frac{N-2}{N+2}}$, Indiana Univ. Math. J., 50 (2001), 611-628. doi: 10.1512/iumj.2001.50.1876.
[25]	J. L. Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl.(9), 71 (1992), 503-526.
[26]	J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Oxford Lecture Series in Mathematics and its Applications 33, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001.
[27]	J. L. Vázquez and M. Winkler, The evolution of singularities in fast diffusion equations: Infinite time blow-down, SIAM J. Math. Anal., 43 (2011), 1499-1535. doi: 10.1137/100809465.
[28]	R. Ye, Global existence and convergence of Yamabe flow, J. Differential Geom., 39 (1994), 35-50. doi: 10.4310/jdg/1214454674.

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