(I am reading a paper called Shortening Complete Plane Curves by Kai-Seng Chou & Xi-Ping Zhu. It is available at http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.jdg/1214424967. Page 476 is relevant)
Consider the partial differential equation on the domain $\mathbb{R}\times [0,T]$: $u_t - A(x,t)u_{xx} + \text{l.o.t} = f$ where $A(x,t) = \frac{(1-k_0(x)^2t)^2}{[(1-k_0(x)^2t)^2 + (k_{0_x}(x)t)^2]^2}$ where $k_0(x)$ is the curvature of the curve $\gamma_0:\mathbb{R} \to \mathbb{R}^2$.
Recall that a PDE is uniformly parabolic if there exist positive constants $a$ and $b$ such that the term in the front of the Laplacian sits between $a$ and $b$, i.e., $a \leq A(x,t) \leq b$.
Questions:
1) Why is it true that
$A$ is bounded in $C^{k, \alpha}(\mathbb{R} \times [0,T])$ if $\gamma_0 \in C^{k+4, \alpha}(\mathbb{R})$?
Something to do with the fact that $k_0$ depends on $(\gamma_0)_{xx}$?
2) Why is it true that
If we restrict $T$ so that, for example, $T < \frac{1}{2}\inf_x \frac{1}{1+k_0^2(x)},$ then the PDE is uniformly parabolic.
I don't see where that comes from at all.
Thanks for any help.