## Differential and Integral Equations

- Differential Integral Equations
- Volume 14, Number 3 (2001), 305-320.

### Global existence and uniqueness of solutions of the Ricci flow equation

#### Abstract

In this paper we will prove the global existence and uniqueness of solutions of the Ricci flow equation on $R^2$ $u_t=\Delta \text{ log }u$, $u>0$, in $R^2\times (0,\infty)$, $u(x,0) =u_0(x)$ for $x\in R^2$, satisfying the inequality $u_t\le u/t$ in $R^2\times (0,\infty)$ and the condition $\liminf_{r\to\infty}$ log $u(x,t)/\text{log }r\ge -2$ uniformly on any compact subset of $(0,\infty)$ as $r=|x|\to\infty$ for any $u_0\not\in L^1(R^2)$, $u_0\ge 0$, satisfying $u_0\in L_{loc}^p(R^2)$ for some $p>1$.

#### Article information

**Source**

Differential Integral Equations, Volume 14, Number 3 (2001), 305-320.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356123330

**Mathematical Reviews number (MathSciNet)**

MR1799897

**Zentralblatt MATH identifier**

1011.35085

**Subjects**

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

Secondary: 35A05 35B25: Singular perturbations 35Kxx: Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35]

#### Citation

Hsu, Shu-Yu. Global existence and uniqueness of solutions of the Ricci flow equation. Differential Integral Equations 14 (2001), no. 3, 305--320. https://projecteuclid.org/euclid.die/1356123330