I'm currently in the proces of learning and writing a bit about the Ricci flow. In particular I'm studying the case of compact 2d Riemannian manifolds. Mostly I'm making good progress but I do miss some background on PDE's. For my text to be complete I need a good reference for the following result:
On a compact Riemannian manifold we have short time existence and uniqueness for the following PDE:
$ \frac{\partial u(x,t)}{\partial t} = \Delta_0 \log u(x,t) + s u(x,t) - S_0(x) = \frac{1}{u}\Delta_0 u(x,t) - \frac{1}{u^2} \vert \nabla u \vert^2 - s u(x,t) - S_0(x,t) $
Here $\Delta_0$ is the Laplace-Beltrami operator with respect to a fixed metric. $s$ is a constant and $S_0$ is the scalar curvature with respect to this fixed metric.
So far I've found one reference in the book "some nonlinear problems in riemannian geometry" by Aubin however this book is very brief about the result does not give a proof.
Do you know a better reference?
Thanks in advance!