    January  2018, 38(1): 91-129. doi: 10.3934/dcds.2018005

## Existence and properties of ancient solutions of the Yamabe flow

 Department of Mathematics, National Chung Cheng University, 168 University Road, Min-Hsiung, Chia-Yi 621, Taiwan

Received  February 2017 Revised  July 2017 Published  September 2017

Let $n≥ 3$ and $m=\frac{n-2}{n+2}$. We construct $5$-parameters, $4$-parameters, and $3$-parameters ancient solutions of the equation $v_t=(v^m)_{xx}+v-v^m$, $v>0$, in $\mathbb{R}× (-∞, T)$ for some $T∈\mathbb{R}$. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient solutions of this equation including exact decay rate of ancient solutions as $|x|\to∞$. We also prove that both the $3$-parameters ancient solution and the $4$-parameters ancient solution are singular limit solution of the $5$-parameters ancient solutions. We also prove the uniqueness of the $4$-parameters ancient solutions. As a consequence we prove that the $4$-parameters ancient solutions that we construct coincide with the $4$-parameters ancient solutions constructed by P. Daskalopoulos, M. del Pino, J. King, and N. Sesum in .

Citation: Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005
##### References:
  S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.  doi: 10.1515/crll.1988.390.79.   S. Brendle, Convergence of the Yamabe flow for arbitrary energy, J. Differential Geom., 69 (2005), 217-278.  doi: 10.4310/jdg/1121449107.   S. Brendle, Convergence of the Yamabe flow in dimension 6 and higher, Invent. Math., 170 (2007), 541-576.  doi: 10.1007/s00222-007-0074-x.   X. Y. Chen and P. Poláčik, Asymptotic periodicity of positive solutions of a diffusion equation on a ball, J. Reine Angew. Math., 472 (1996), 17-51. B. E. J. Dahlberg and C. Kenig, Nonnegative solutions of the generalized porous medium equations, Revista Matemática Iberoamericana, 2 (1986), 267-305.  P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, arXiv: 1306. 0859v1. P. Daskalopoulos, M. del Pino, J. King and N. Sesum, Type Ⅰ ancient compact solutions of the Yamabe flow, Nonlinear Analysis, Theory, Methods and Applications, 137 (2016), 338-356.  doi: 10.1016/j.na.2015.12.005.   P. Daskalopoulos, M. del Pino, J. King and N. Sesum, New type Ⅰ ancient compact solutions of the Yamabe flow, arXiv: 1601. 05349v1. F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure and Applied Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.   S. Y. Hsu, Asymptotic behaviour of solutions of the equation $u_t=Δ\log u$ near the extinction time, Adv. Differential Equations, 8 (2003), 161-187.  S. Y. Hsu, Singular limit and exact decay rate of a nonlinear elliptic equation, Nonlinear Analysis TMA, 75 (2012), 3443-3455.  doi: 10.1016/j.na.2012.01.009.   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.   K. M. Hui, Existence of solutions of the equation $u_t=Δ\log u$, Nonlinear Analysis TMA, 37 (1999), 875-914.  doi: 10.1016/S0362-546X(98)00081-9.   O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Mono. , Amer. Math. Soc. , Providence, R. I. , USA, 1968.  H. Matano, Nonincrease of the lap number of a solution for one dimensional semi-linear parabolic, equations, J. Fac. Sci. Univ. Tokyo, Sec., 29 (1982), 401-441.  A. de Pablo and J. L. Vazquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equations, 93 (1991), 19-61.  doi: 10.1016/0022-0396(91)90021-Z.   M. del Pino and M. Sáez, On the extinction profile for solutions of $u_t=Δ u^{(N-2)/(N+2)}$, Indiana Univ. Math. J., 50 (2001), 611-628.  doi: 10.1512/iumj.2001.50.1876.   A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, Blow-up in Quasilinear Parabolic Equations Walter de Gruyter, Berlin, 1995. doi: 10.1515/9783110889864.  show all references

##### References:
  S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.  doi: 10.1515/crll.1988.390.79.   S. Brendle, Convergence of the Yamabe flow for arbitrary energy, J. Differential Geom., 69 (2005), 217-278.  doi: 10.4310/jdg/1121449107.   S. Brendle, Convergence of the Yamabe flow in dimension 6 and higher, Invent. Math., 170 (2007), 541-576.  doi: 10.1007/s00222-007-0074-x.   X. Y. Chen and P. Poláčik, Asymptotic periodicity of positive solutions of a diffusion equation on a ball, J. Reine Angew. Math., 472 (1996), 17-51. B. E. J. Dahlberg and C. Kenig, Nonnegative solutions of the generalized porous medium equations, Revista Matemática Iberoamericana, 2 (1986), 267-305.  P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, arXiv: 1306. 0859v1. P. Daskalopoulos, M. del Pino, J. King and N. Sesum, Type Ⅰ ancient compact solutions of the Yamabe flow, Nonlinear Analysis, Theory, Methods and Applications, 137 (2016), 338-356.  doi: 10.1016/j.na.2015.12.005.   P. Daskalopoulos, M. del Pino, J. King and N. Sesum, New type Ⅰ ancient compact solutions of the Yamabe flow, arXiv: 1601. 05349v1. F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure and Applied Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.   S. Y. Hsu, Asymptotic behaviour of solutions of the equation $u_t=Δ\log u$ near the extinction time, Adv. Differential Equations, 8 (2003), 161-187.  S. Y. Hsu, Singular limit and exact decay rate of a nonlinear elliptic equation, Nonlinear Analysis TMA, 75 (2012), 3443-3455.  doi: 10.1016/j.na.2012.01.009.   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.   K. M. Hui, Existence of solutions of the equation $u_t=Δ\log u$, Nonlinear Analysis TMA, 37 (1999), 875-914.  doi: 10.1016/S0362-546X(98)00081-9.   O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Mono. , Amer. Math. Soc. , Providence, R. I. , USA, 1968.  H. Matano, Nonincrease of the lap number of a solution for one dimensional semi-linear parabolic, equations, J. Fac. Sci. Univ. Tokyo, Sec., 29 (1982), 401-441.  A. de Pablo and J. L. Vazquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equations, 93 (1991), 19-61.  doi: 10.1016/0022-0396(91)90021-Z.   M. del Pino and M. Sáez, On the extinction profile for solutions of $u_t=Δ u^{(N-2)/(N+2)}$, Indiana Univ. Math. J., 50 (2001), 611-628.  doi: 10.1512/iumj.2001.50.1876.   A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, Blow-up in Quasilinear Parabolic Equations Walter de Gruyter, Berlin, 1995. doi: 10.1515/9783110889864.  Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613  Luiz F. O. Faria. Existence and uniqueness of positive solutions for singular biharmonic elliptic systems. Conference Publications, 2015, 2015 (special) : 400-408. doi: 10.3934/proc.2015.0400  Yen-Lin Wu, Zhi-You Chen, Jann-Long Chern, Y. Kabeya. Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. Communications on Pure and Applied Analysis, 2014, 13 (2) : 949-960. doi: 10.3934/cpaa.2014.13.949  Taebeom Kim, Sunčica Čanić, Giovanna Guidoboni. Existence and uniqueness of a solution to a three-dimensional axially symmetric Biot problem arising in modeling blood flow. 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