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. DaskalopoulosM. del PinoJ. 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. DaskalopoulosM. del PinoJ. 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

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