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Regularity of elliptic systems in divergence form with directional homogenization
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 |
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 [
References:
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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. |
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. |
| [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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