I have the following PDE $u_t = a(x,t)u_{xx} + b(x,t)u_x + c(x,t)u - f(x,t)$ $u|_{t=0} = u_0$ over the domain $S^1 \times [0,T)$. The coefficients and $f$ are in $C^{k,\alpha}$ for some $k$ (and are $2\pi$ periodic, ignore if this doesn't make sense). Also $a$ is such that the equation is uniformly parabolic.
My questions: 1) To get an a-priori estimate for this equation
$\lVert u \rVert_{C^{k+2, \alpha}} \leq C(\lVert f \rVert_{C^{k, \alpha}} + \lVert u_0 \rVert_{C^{k+2, \alpha}})$
what do I do? The only thing I know of is multiplying by a test function and integrating and using Gronwall but this gives me norms in Sobolev spaces, I believe.
2) Apparently, the following is true, but I need some explanation:
There is a unique solution $u \in C^{k+2, \alpha}$. Proof: first solve the Cauchy problem in the smooth category by means of separation of variables. Then use an approximation argument coupling with the global a-priori estimate above to get the general result.
What's the Cauchy problem (Wikipedia doesn't help. It just says the domain is a manifold)? What's smooth category? What's the approximation argument thing? Sorry if these questions are stupid but I have never heard of this stuff.
Any references or help would be appreciated.