How to solve this quasilinear parabolic evolution equation (result of curve shortening flow)? In the classic paper by Hamilton and Gage (see http://intlpress.com/JDG/archive/1986/23-1-69.pdf), they give the PDE problem:

Find $k:S^1 \times [0,T) \to \mathbb{R} \text{ s.t }$

$$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\text{(i) } k \in C^{2+\alpha, 1+\alpha}(S^1 \times [0,T-\epsilon]) \text{ for all } \epsilon &gt; 0$$

$$\text{(ii) }\frac{\partial k}{\partial t} = k^2 \frac{\partial^2 k}{\partial \theta^2} + k^3$$

$$\;\;\;\;\;\;\;\;\;\text{(iii) }k(\theta, 0) = \psi(\theta) \text{ satisfies }$$

$$\qquad \;\;\;\;\;\;\text{(a) } \psi \in C^{1+\alpha(S^1)}$$

$$\qquad \text{(b) } \psi(\theta) &gt; 0$$

$$\qquad \;\;\;\;\;\;\;\;\;\;\;\;\text{(c) } \int_0^{2\pi} \frac{\cos(\theta)}{\psi(\theta))} = 0$$

And they say that is a shown from standard results. Every paper that I've seen that references this paper says that H-G prove this (but they don't, they just state it), and the ones that talk about it give references to papers that are difficult (maximal regularity and Volterra conditions and I don't know which spaces to use as it's quite an abstract result (by Herbert Amann)) that may not be necessary to use anyway. Does anyone know what the standard results are? I have a book on quasilinear equations but it looks like this equation doesn't fit the conditions that are required for the existence result it gives. It's all very confusing because there are lots of different requisites in different sources...

I appreciate any help.
Gage and Hamilton prove this fact in the paper "The Heat Equation Shrinking Convex Plane Curves" see page 80 of this paper.. I confess that I didn't understand the argument too.. 
I can't access the link you furnished..
Gage and Hamilton prove this fact in the paper "The Heat Equation Shrinking Convex Plane Curves" see page 80 of this paper.. I confess that I didn't understand the argument too.. 
I can't access the link you furnished..