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

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

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 $τ\to∞$.
Citation: Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

A. BlanchetM. BonforteJ. DolbeaultG. 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.  Google Scholar

[3]

M. BonforteJ. DolbeaultG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, arXiv: 1306.0859. Google Scholar

[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.  Google Scholar

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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

[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.  Google Scholar

[10]

M. FilaJ. L. VázquezM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

K. M. Hui, On some Dirichlet and Cauchy problems for a singular diffusion equation, Differential Integral Equations, 15 (2002), 769-804.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss. 132, Springer-Verlag, Berlin, New York, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. L. Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl.(9), 71 (1992), 503-526.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

R. Ye, Global existence and convergence of Yamabe flow, J. Differential Geom., 39 (1994), 35-50.  doi: 10.4310/jdg/1214454674.  Google Scholar

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.  Google Scholar

[2]

A. BlanchetM. BonforteJ. DolbeaultG. 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.  Google Scholar

[3]

M. BonforteJ. DolbeaultG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, arXiv: 1306.0859. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. FilaJ. L. VázquezM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

K. M. Hui, On some Dirichlet and Cauchy problems for a singular diffusion equation, Differential Integral Equations, 15 (2002), 769-804.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss. 132, Springer-Verlag, Berlin, New York, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. L. Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl.(9), 71 (1992), 503-526.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

R. Ye, Global existence and convergence of Yamabe flow, J. Differential Geom., 39 (1994), 35-50.  doi: 10.4310/jdg/1214454674.  Google Scholar

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