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On trigonometric skew-products over irrational circle-rotations
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 |
$ n\ge 3 $ |
$ 0<m<\frac{n-2}{n} $ |
$ \beta<0 $ |
$ \alpha = \frac{2\beta}{1-m} $ |
$ (\mathbb{R}^n\setminus\{0\})\times \mathbb{R} $ |
$ U_{\lambda}(x,t) = e^{-\alpha t}f_{\lambda}(e^{-\beta t}x), x\in \mathbb{R}^n\setminus\{0\}, t\in\mathbb{R}, $ |
$ f_{\lambda} $ |
$ \frac{n-1}{m}\Delta f^m+\alpha f+\beta x\cdot\nabla f = 0 \text{ in }\mathbb{R}^n\setminus\{0\}, $ |
$ \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)} $ |
$ \underset{\substack{r\to\infty}}{\lim}r^{\frac{n-2}{m}}f(r) = \lambda^{\frac{2}{1-m}-\frac{n-2}{m}} $ |
$ \lambda>0 $ |
$ u_t = \frac{n-1}{m}\Delta u^m $ |
$ (\mathbb{R}^n\setminus\{0\})\times (0,\infty) $ |
$ u_0 $ |
$ f_{\lambda_1}(x)\le u_0(x)\le f_{\lambda_2}(x) $ |
$ \forall x\in\mathbb{R}^n\setminus\{0\} $ |
$ u $ |
$ 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 $ |
$ \lambda_1>\lambda_2>0 $ |
$ u $ |
$ n = 3,4 $ |
$ \frac{n-2}{n+2}\le m<\frac{n-2}{n} $ |
$ u $ |
$ 3\le n<8 $ |
$ 1-\sqrt{2/n}\le m<\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right) $ |
$ u(x,t) $ |
$ x\in\mathbb{R}^n\setminus\{0\} $ |
$ t>0 $ |
$ u_0 $ |
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. 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. |
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