# 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 & Continuous Dynamical Systems - A, 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. Google Scholar [2] S. Brendle, Convergence of the Yamabe flow for arbitrary energy, J. Differential Geom., 69 (2005), 217-278. doi: 10.4310/jdg/1121449107. Google Scholar [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. Google Scholar [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. Google Scholar [5] B. E. J. Dahlberg and C. Kenig, Nonnegative solutions of the generalized porous medium equations, Revista Matemática Iberoamericana, 2 (1986), 267-305. Google Scholar [6] P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, arXiv: 1306. 0859v1.Google Scholar [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. Google Scholar [8] P. Daskalopoulos, M. del Pino, J. King and N. Sesum, New type Ⅰ ancient compact solutions of the Yamabe flow, arXiv: 1601. 05349v1.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

show all references

##### 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. Google Scholar [2] S. Brendle, Convergence of the Yamabe flow for arbitrary energy, J. Differential Geom., 69 (2005), 217-278. doi: 10.4310/jdg/1121449107. Google Scholar [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. Google Scholar [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. Google Scholar [5] B. E. J. Dahlberg and C. Kenig, Nonnegative solutions of the generalized porous medium equations, Revista Matemática Iberoamericana, 2 (1986), 267-305. Google Scholar [6] P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, arXiv: 1306. 0859v1.Google Scholar [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. Google Scholar [8] P. Daskalopoulos, M. del Pino, J. King and N. Sesum, New type Ⅰ ancient compact solutions of the Yamabe flow, arXiv: 1601. 05349v1.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
 [1] Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613 [2] 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 [3] 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 & Applied Analysis, 2014, 13 (2) : 949-960. doi: 10.3934/cpaa.2014.13.949 [4] L. Brandolini, M. Rigoli and A. G. Setti. On the existence of positive solutions of Yamabe-type equations on the Heisenberg group. Electronic Research Announcements, 1996, 2: 101-107. [5] 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. Communications on Pure & Applied Analysis, 2010, 9 (4) : 839-865. doi: 10.3934/cpaa.2010.9.839 [6] Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267 [7] Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463 [8] Rui Peng, Xianfa Song, Lei Wei. Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4489-4505. doi: 10.3934/dcds.2017192 [9] Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213 [10] G. C. Yang, K. Q. Lan. Systems of singular integral equations and applications to existence of reversed flow solutions of Falkner-Skan equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2465-2495. doi: 10.3934/cpaa.2013.12.2465 [11] Maria Assunta Pozio, Fabio Punzo, Alberto Tesei. Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 891-916. doi: 10.3934/dcds.2011.30.891 [12] Jann-Long Chern, Yong-Li Tang, Chuan-Jen Chyan, Yi-Jung Chen. On the uniqueness of singular solutions for a Hardy-Sobolev equation. Conference Publications, 2013, 2013 (special) : 123-128. doi: 10.3934/proc.2013.2013.123 [13] Maria Assunta Pozio, Alberto Tesei. On the uniqueness of bounded solutions to singular parabolic problems. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 117-137. doi: 10.3934/dcds.2005.13.117 [14] Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 1-8. doi: 10.3934/nhm.2013.8.1 [15] Patricia J.Y. Wong. Existence of solutions to singular integral equations. Conference Publications, 2009, 2009 (Special) : 818-827. doi: 10.3934/proc.2009.2009.818 [16] Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047 [17] Chunqing Lu. Existence and uniqueness of single spike solution of the carrier-pearson problem. Conference Publications, 2001, 2001 (Special) : 259-264. doi: 10.3934/proc.2001.2001.259 [18] Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic & Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707 [19] Dominique Blanchard, Olivier Guibé, Hicham Redwane. Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (1) : 197-217. doi: 10.3934/cpaa.2016.15.197 [20] Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033

2018 Impact Factor: 1.143