Parabolic linear PDE with continuous coefficients; how to solve and explanation of text needed 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.
 A: As Andrew commented, obtaining Hölder estimates can sometimes be labour intensive. So I'll ignore part (1) for now. 
For part (2):


*

*The term "Cauchy problem" is just another name for "initial(-boundary) value problem". In your case there is no boundary since the spatial domain is a closed manifold. 

*Smooth category just means that given smooth initial data, try to solve the equation with smooth solution. Usually this is obtained by separation of variables or explicitly integrating against some Green's function. 

*Approximation argument is the idea that for most reasonable function (Banach) spaces $X$ defined on a manifold $M$, the space $C^\infty_0(M)$ of smooth functions with compact support is dense in $X$ in the Banach norm. Hence we can approximate a, say, $C^{k,\alpha}$ initial data with a sequence of $C^\infty$ initial data. Now if there is a good a priori estimate available (such as the Hölder estimate you quoted in your question), then we can transfer the convergence of the initial data to a convergence of the sequence of smooth solutions in $C^{k+2,\alpha}$ norm, obtaining then a $C^{k+2,\alpha}$ solution as the limit of a sequence fo smooth solutions, which we derived from the sequence of smooth initial data  approximating the $C^{k,\alpha}$ initial data. 

