I am doing a work on Curve Shortening Flow. I want to prove some theorems that I use to deduce the existence of solution and a maximum principle to check that all the curves of the solution are convex. What I have is $$ u_t= \frac{1}{v^2}u_{xx} $$ where $v=\langle u_x,u_x \rangle$ and $$ u: S^1×[0,T) \to \mathbb{R}^2 $$ with $u(\cdot,t): S^1 \to \mathbb{R}^2$. The other important PDE is for the curvature of the curves, which satisfies $$k_t=k_{ss}+k^3$$ $s$ the arc-length parameter. The theorems for short time existence or maximal solution are for the first PDE and the maximum principle is for the second. I need help to find books where I can find proves to that theorems. Do you know any useful book?
Thank you in advance.