Solvability of Perelman's $\mathcal W$ system.

Question

How to show the system have solution ? $R_{ij}$ is ricci tensor, $R$ is scalar curvature. I feel this is complex question, because I have little knowledge about PDE. So, if it is complex, just tell me what I should read ? I try to find answer in Evans' PDE book, but I am not sure. Whether the 11 chapter of Evans' PDE can resolve this question ?

The below system is from 206 page of this paper.

\begin{cases} \frac{\partial g_{ij}}{\partial t}=-2R_{ij} \\ \frac{\partial f}{\partial t}=-\Delta f+|\nabla f|^2-R+\frac{n}{2\tau} \\ \frac{\partial \tau}{\partial t}=1 \end{cases}

I try it. Under some initial condition. I can get $\tau=\tau_0+t$. Besides, $\frac{\partial g_{ij}}{\partial t}=-2R_{ij}$ has short-time solution, according to the 1.2 of this. So, it just be whether $\frac{\partial f}{\partial t}=-\Delta f+|\nabla f|^2-R(t)+\frac{n}{2(\tau_0+t)}$ has solution. Because $\tau_0 >0,t\ge0$,and $R(t)$ is smooth...

Answer 1

There is an answer about your question.Let $\tau=t_0-t$, $u=e^{-f}$, then the nonlinear equation will become a linear equation.