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

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 $
.
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. DaskalopoulosJ. 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. DaskalopoulosM. 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. 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

[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. DaskalopoulosJ. 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. DaskalopoulosM. 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. 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

[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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