Existence and properties of ancient solutions of the Yamabe flow

Shu-Yu Hsu

doi:10.3934/dcds.2018005

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2018, Volume 38, Issue 1: 91-129. Doi: 10.3934/dcds.2018005

This issue Previous Article Regularity of elliptic systems in divergence form with directional homogenization Next Article Invariant curves of smooth quasi-periodic mappings

Existence and properties of ancient solutions of the Yamabe flow

Shu-Yu Hsu

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

Received: January 31, 2017

Revised: June 30, 2017

Published: October 2017

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Abstract

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].

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Mathematics Subject Classification: Primary:35K55, 53C44;Secondary:35A01, 35B44.

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