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November  2021, 41(11): 5473-5508. doi: 10.3934/dcds.2021085

Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space

Kin Ming Hui 1, and  Jinwan Park 2,,

1. 

Institute of Mathematics, Academia Sinica, Taipei, Taiwan, R. O. C
2. 

Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

* Corresponding author: Jinwan Park

Received  July 2020 Revised  April 2021 Published  November 2021 Early access  May 2021

Fund Project: The corresponding author is supported by NRF grant 2020R1A6A3A01099425

For
$ n\ge 3 $
,
$ 0<m<\frac{n-2}{n} $
,
$ \beta<0 $
 and
$ \alpha = \frac{2\beta}{1-m} $
, we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in
$ (\mathbb{R}^n\setminus\{0\})\times \mathbb{R} $
 of the form
$ U_{\lambda}(x,t) = e^{-\alpha t}f_{\lambda}(e^{-\beta t}x), x\in \mathbb{R}^n\setminus\{0\}, t\in\mathbb{R}, $
 where
$ f_{\lambda} $
 is a radially symmetric function satisfying
$ \frac{n-1}{m}\Delta f^m+\alpha f+\beta x\cdot\nabla f = 0 \text{ in }\mathbb{R}^n\setminus\{0\}, $
with
$ \underset{\substack{r\to 0}}{\lim}\frac{r^2f(r)^{1-m}}{\log r^{-1}} = \frac{2(n-1)(n-2-nm)}{|\beta|(1-m)} $
 and
$ \underset{\substack{r\to\infty}}{\lim}r^{\frac{n-2}{m}}f(r) = \lambda^{\frac{2}{1-m}-\frac{n-2}{m}} $
, for some constant
$ \lambda>0 $
.
As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation
$ u_t = \frac{n-1}{m}\Delta u^m $
 in
$ (\mathbb{R}^n\setminus\{0\})\times (0,\infty) $
 with initial value
$ u_0 $
 satisfying
$ f_{\lambda_1}(x)\le u_0(x)\le f_{\lambda_2}(x) $
,
$ \forall x\in\mathbb{R}^n\setminus\{0\} $
, such that the solution
$ u $
 satisfies
$ U_{\lambda_1}(x,t)\le u(x,t)\le U_{\lambda_2}(x,t) $
,
$ \forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0 $
, for some constants
$ \lambda_1>\lambda_2>0 $
.
We also prove the asymptotic large time behaviour of such singular solution
$ u $
 when
$ n = 3,4 $
 and
$ \frac{n-2}{n+2}\le m<\frac{n-2}{n} $
 holds. Asymptotic large time behaviour of such singular solution
$ u $
 is also obtained when
$ 3\le n<8 $
,
$ 1-\sqrt{2/n}\le m<\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right) $
, and
$ u(x,t) $
 is radially symmetric in
$ x\in\mathbb{R}^n\setminus\{0\} $
 for any
$ t>0 $
 under appropriate conditions on the initial value
$ u_0 $
.
Keywords: asymptotic behaviour of solutions, blow-up, fast diffusion equation, singular solution, L1-contraction, radially symmetric self-similar solution.
Mathematics Subject Classification: Primary: 35B40, 35B44, 35K55, 35K67.
Citation: Kin Ming Hui, Jinwan Park. Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5473-5508. doi: 10.3934/dcds.2021085
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]

B. Choi and P. Daskalopoulos, Yamabe flow: Steady solitons and type Ⅱ singularities, Nonlinear Analysis, 173 (2018), 1-18.  doi: 10.1016/j.na.2018.03.008.  Google Scholar

[3]

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

[4]

P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, Comm. Anal. Geom., 27 (2019), 1757-1798.  doi: 10.4310/CAG.2019.v27.n8.a4.  Google Scholar

[5]

P. Daskalopoulos, M. del Pino and N. Sesum, Type Ⅱ ancient compact solutions to the Yamabe flow, J. Reine Angew. Math., 738 (2018), 1-71.  doi: 10.1515/crelle-2015-0048.  Google Scholar

[6]

P. Daskalopoulos and N. Sesum, On the extinction profile of solutions to fast diffusion, J. Reine Angew. Math., 2008 (2008), 95-119.  doi: 10.1515/CRELLE.2008.066.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

M. Fila and M. Winkler, Slow growth of solutions of superfast diffusion equations with unbounded initial data, J. London Math. Soc., 95 (2017), 659-683.  doi: 10.1112/jlms.12029.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

S. Y. Hsu, Some properties of the Yamabe soliton and the related nonlinear elliptic equation, Calc. Var. Partial Differential Equations, 49 (2014), 307-321.  doi: 10.1007/s00526-012-0583-3.  Google Scholar

[16]

S. Y. Hsu, Global behaviour of solutions of the fast diffusion equation, Manuscripta Math., 158 (2019), 103-117.  doi: 10.1007/s00229-018-1008-1.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

K. M. Hui and S. Kim, Asymptotic large time behavior of singular solutions of the fast diffusion equation, Discrete Contin. Dyn. Syst., 37 (2017), 5943-5977.  doi: 10.3934/dcds.2017258.  Google Scholar

[21]

K. M. Hui and S. Kim, Existence and large time behaviour of finite points blow-up solutions of the fast diffusion equation, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 112, 39pp. doi: 10.1007/s00526-018-1396-9.  Google Scholar

[22]

T. Jin and J. Xiong, Singular extinction profiles of solutions to some fast diffusion equations, preprint, arXiv: 2008.02059. Google Scholar

[23]

T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148.  doi: 10.1007/BF02760233.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

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]

B. Choi and P. Daskalopoulos, Yamabe flow: Steady solitons and type Ⅱ singularities, Nonlinear Analysis, 173 (2018), 1-18.  doi: 10.1016/j.na.2018.03.008.  Google Scholar

[3]

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

[4]

P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, Comm. Anal. Geom., 27 (2019), 1757-1798.  doi: 10.4310/CAG.2019.v27.n8.a4.  Google Scholar

[5]

P. Daskalopoulos, M. del Pino and N. Sesum, Type Ⅱ ancient compact solutions to the Yamabe flow, J. Reine Angew. Math., 738 (2018), 1-71.  doi: 10.1515/crelle-2015-0048.  Google Scholar

[6]

P. Daskalopoulos and N. Sesum, On the extinction profile of solutions to fast diffusion, J. Reine Angew. Math., 2008 (2008), 95-119.  doi: 10.1515/CRELLE.2008.066.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

M. Fila and M. Winkler, Slow growth of solutions of superfast diffusion equations with unbounded initial data, J. London Math. Soc., 95 (2017), 659-683.  doi: 10.1112/jlms.12029.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

S. Y. Hsu, Some properties of the Yamabe soliton and the related nonlinear elliptic equation, Calc. Var. Partial Differential Equations, 49 (2014), 307-321.  doi: 10.1007/s00526-012-0583-3.  Google Scholar

[16]

S. Y. Hsu, Global behaviour of solutions of the fast diffusion equation, Manuscripta Math., 158 (2019), 103-117.  doi: 10.1007/s00229-018-1008-1.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

K. M. Hui and S. Kim, Asymptotic large time behavior of singular solutions of the fast diffusion equation, Discrete Contin. Dyn. Syst., 37 (2017), 5943-5977.  doi: 10.3934/dcds.2017258.  Google Scholar

[21]

K. M. Hui and S. Kim, Existence and large time behaviour of finite points blow-up solutions of the fast diffusion equation, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 112, 39pp. doi: 10.1007/s00526-018-1396-9.  Google Scholar

[22]

T. Jin and J. Xiong, Singular extinction profiles of solutions to some fast diffusion equations, preprint, arXiv: 2008.02059. Google Scholar

[23]

T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148.  doi: 10.1007/BF02760233.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

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