# American Institute of Mathematical Sciences

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 [8].

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:
 [1] 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. [2] S. Brendle, Convergence of the Yamabe flow for arbitrary energy, J. Differential Geom., 69 (2005), 217-278.  doi: 10.4310/jdg/1121449107. [3] 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. [4] 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. [5] B. E. J. Dahlberg and C. Kenig, Nonnegative solutions of the generalized porous medium equations, Revista Matemática Iberoamericana, 2 (1986), 267-305. [6] P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, arXiv: 1306. 0859v1. [7] 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. [8] P. Daskalopoulos, M. del Pino, J. King and N. Sesum, New type Ⅰ ancient compact solutions of the Yamabe flow, arXiv: 1601. 05349v1. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] 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. [15] 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. [16] 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. [17] 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. [18] 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.

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##### References:
 [1] 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. [2] S. Brendle, Convergence of the Yamabe flow for arbitrary energy, J. Differential Geom., 69 (2005), 217-278.  doi: 10.4310/jdg/1121449107. [3] 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. [4] 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. [5] B. E. J. Dahlberg and C. Kenig, Nonnegative solutions of the generalized porous medium equations, Revista Matemática Iberoamericana, 2 (1986), 267-305. [6] P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, arXiv: 1306. 0859v1. [7] 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. [8] P. Daskalopoulos, M. del Pino, J. King and N. Sesum, New type Ⅰ ancient compact solutions of the Yamabe flow, arXiv: 1601. 05349v1. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] 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. [15] 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. [16] 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. [17] 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. [18] 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.
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