Existence and properties of ancient solutions of the Yamabe flow

TL;DR

This paper constructs and analyzes various ancient solutions to a nonlinear PDE related to the Yamabe flow, revealing their properties, decay rates, and relationships as limits of higher-parameter solutions.

Contribution

It introduces explicit multi-parameter ancient solutions for the Yamabe flow equation and studies their decay properties and limiting behaviors.

Findings

Constructed 3, 4, and 5-parameter ancient solutions.

Determined exact decay rates as |x|→∞.

Showed 3- and 4-parameter solutions as limits of 5-parameter solutions.

Abstract

Let $n\ge 3$ and $m=\frac{n-2}{n+2}$. We construct $5$-parameters, $4$-parameters, $3$-parameters ancient solutions of the equation $v_t=(v^m)_{xx}+v-v^m$, $v>0$, in $\mathbb{R}\times (-\infty,T)$ for some $T\in\mathbb{R}$. 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 $|x|\to\infty$. 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.