# American Institute of Mathematical Sciences

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 , $n≥3$. 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. 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. 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. 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. 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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. 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. 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. 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. 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. 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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