Solutions with snaking singularities for the fast diffusion equation

Fila, Marek; King, John; Takahashi, Jin; Yanagida, Eiji

doi:10.1090/tran/8479

Solutions with snaking singularities for the fast diffusion equation
HTML articles powered by AMS MathViewer

by Marek Fila, John Robert King, Jin Takahashi and Eiji Yanagida PDF

Trans. Amer. Math. Soc. 374 (2021), 8775-8792 Request permission

Abstract:

We construct solutions of the fast diffusion equation, which exist for all $t\in \mathbb {R}$ and are singular on the set $\Gamma (t)≔\{ \xi (s) ; -\infty <s \leq ct \}$, $c>0$, where $\xi \in C^3(\mathbb {R};\mathbb {R}^n)$, $n\geq 2$. We also give a precise description of the behavior of the solutions near $\Gamma (t)$.

References

E. Chasseigne, Classification of razor blades to the filtration equation—the sublinear case, J. Differential Equations 187 (2003), no. 1, 72–105. MR 1946547, DOI 10.1016/S0022-0396(02)00019-0
Emmanuel Chasseigne and Juan Luis Vazquez, Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities, Arch. Ration. Mech. Anal. 164 (2002), no. 2, 133–187. MR 1929929, DOI 10.1007/s00205-002-0210-0
Emmanuel Chasseigne and Juan Luis Vázquez, The pressure equation in the fast diffusion range, Rev. Mat. Iberoamericana 19 (2003), no. 3, 873–917. MR 2053567, DOI 10.4171/RMI/373
Xinfu Chen and Jong-Shenq Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations 212 (2005), no. 1, 62–84. MR 2130547, DOI 10.1016/j.jde.2004.10.028
Marek Fila, Petra Macková, Jin Takahashi, and Eiji Yanagida, Moving singularities for nonlinear diffusion equations in two space dimensions, J. Elliptic Parabol. Equ. 6 (2020), no. 1, 155–169. MR 4114347, DOI 10.1007/s41808-020-00062-0
Marek Fila, Jin Takahashi, and Eiji Yanagida, Solutions with moving singularities for equations of porous medium type, Nonlinear Anal. 179 (2019), 237–253. MR 3886631, DOI 10.1016/j.na.2018.08.016
Yukitaka Fukao, Yoshihisa Morita, and Hirokazu Ninomiya, Some entire solutions of the Allen-Cahn equation, Proceedings of Third East Asia Partial Differential Equation Conference, 2004, pp. 15–32. MR 2057634, DOI 10.11650/twjm/1500558454
Brian H. Gilding and Robert Kersner, Travelling waves in nonlinear diffusion-convection reaction, Progress in Nonlinear Differential Equations and their Applications, vol. 60, Birkhäuser Verlag, Basel, 2004. MR 2081104, DOI 10.1007/978-3-0348-7964-4
Miguel A. Herrero and Michel Pierre, The Cauchy problem for $u_t=\Delta u^m$ when $0<m<1$, Trans. Amer. Math. Soc. 291 (1985), no. 1, 145–158. MR 797051, DOI 10.1090/S0002-9947-1985-0797051-0
Khin Phyu Phyu Htoo, Jin Takahashi, and Eiji Yanagida, Higher-dimensional moving singularities in a superlinear parabolic equation, J. Evol. Equ. 18 (2018), no. 4, 1575–1593. MR 3884761, DOI 10.1007/s00028-018-0452-4
Khin Phyu Phyu Htoo and Eiji Yanagida, Singular solutions of a superlinear parabolic equation with homogeneous Neumann boundary conditions, Nonlinear Anal. 151 (2017), 96–108. MR 3596672, DOI 10.1016/j.na.2016.11.015
Kin Ming Hui and Soojung Kim, Asymptotic large time behavior of singular solutions of the fast diffusion equation, Discrete Contin. Dyn. Syst. 37 (2017), no. 11, 5943–5977. MR 3681967, DOI 10.3934/dcds.2017258
Toru Kan and Jin Takahashi, On the profile of solutions with time-dependent singularities for the heat equation, Kodai Math. J. 37 (2014), no. 3, 568–585. MR 3273884, DOI 10.2996/kmj/1414674609
Toru Kan and Jin Takahashi, Time-dependent singularities in semilinear parabolic equations: behavior at the singularities, J. Differential Equations 260 (2016), no. 10, 7278–7319. MR 3473442, DOI 10.1016/j.jde.2016.01.026
Toru Kan and Jin Takahashi, Time-dependent singularities in semilinear parabolic equations: existence of solutions, J. Differential Equations 263 (2017), no. 10, 6384–6426. MR 3693178, DOI 10.1016/j.jde.2017.07.016
Grzegorz Karch, Maria E. Schonbek, and Tomas P. Schonbek, Singularities of certain finite energy solutions to the Navier-Stokes system, Discrete Contin. Dyn. Syst. 40 (2020), no. 1, 189–206. MR 4026957, DOI 10.3934/dcds.2020008
Grzegorz Karch and Xiaoxin Zheng, Time-dependent singularities in the Navier-Stokes system, Discrete Contin. Dyn. Syst. 35 (2015), no. 7, 3039–3057. MR 3343553, DOI 10.3934/dcds.2015.35.3039
J. R. King, Self-similar behaviour for the equation of fast nonlinear diffusion, Philos. Trans. Roy. Soc. London Ser. A 343 (1993), 337–375., DOI 10.1098/rsta.1993.0052
Shota Sato and Eiji Yanagida, Solutions with moving singularities for a semilinear parabolic equation, J. Differential Equations 246 (2009), no. 2, 724–748. MR 2468735, DOI 10.1016/j.jde.2008.09.004
Shota Sato and Eiji Yanagida, Forward self-similar solution with a moving singularity for a semilinear parabolic equation, Discrete Contin. Dyn. Syst. 26 (2010), no. 1, 313–331. MR 2552790, DOI 10.3934/dcds.2010.26.313
Shota Sato and Eiji Yanagida, Singular backward self-similar solutions of a semilinear parabolic equation, Discrete Contin. Dyn. Syst. Ser. S 4 (2011), no. 4, 897–906. MR 2746450, DOI 10.3934/dcdss.2011.4.897
Jin Takahashi, Existence of solutions with moving singularities for a semilinear heat equation with a critical exponent, J. Math. Pures Appl. (9) 148 (2021), 128–149 (English, with English and French summaries). MR 4223350, DOI 10.1016/j.matpur.2021.02.007
Jin Takahashi and Hikaru Yamamoto, Solutions with time-dependent singular sets for the heat equation with absorption, J. Differential Equations 266 (2019), no. 7, 4061–4105. MR 3912711, DOI 10.1016/j.jde.2018.09.029
Jin Takahashi and Eiji Yanagida, Time-dependent singularities in the heat equation, Commun. Pure Appl. Anal. 14 (2015), no. 3, 969–979. MR 3320161, DOI 10.3934/cpaa.2015.14.969
Jin Takahashi and Eiji Yanagida, Time-dependent singularities in a semilinear parabolic equation with absorption, Commun. Contemp. Math. 18 (2016), no. 5, 1550077, 27. MR 3523183, DOI 10.1142/S0219199715500777
Juan Luis Vázquez, Smoothing and decay estimates for nonlinear diffusion equations, Oxford Lecture Series in Mathematics and its Applications, vol. 33, Oxford University Press, Oxford, 2006. Equations of porous medium type. MR 2282669, DOI 10.1093/acprof:oso/9780199202973.001.0001
Juan Luis Vázquez and Michael Winkler, The evolution of singularities in fast diffusion equations: infinite-time blow-down, SIAM J. Math. Anal. 43 (2011), no. 4, 1499–1535. MR 2821594, DOI 10.1137/100809465