# American Institute of Mathematical Sciences

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 , $ \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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] K. M. Hui, On some Dirichlet and Cauchy problems for a singular diffusion equation, Differential Integral Equations, 15 (2002), 769-804. [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. [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. [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. [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. [22] T. Jin and J. Xiong, Singular extinction profiles of solutions to some fast diffusion equations, preprint, arXiv: 2008.02059. [23] T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148. doi: 10.1007/BF02760233. [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. [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. [26] J. L. Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl., 71 (1992), 503-526. [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. [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. 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] 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] K. M. Hui, On some Dirichlet and Cauchy problems for a singular diffusion equation, Differential Integral Equations, 15 (2002), 769-804. [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. [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. [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. [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. [22] T. Jin and J. Xiong, Singular extinction profiles of solutions to some fast diffusion equations, preprint, arXiv: 2008.02059. [23] T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148. doi: 10.1007/BF02760233. [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. [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. 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