Let's consider a PDE of the form (for a function $f$ of variables $t, x_1, \dots, x_n$) $$\frac\partial{\partial t} f = \sum_{ij}C_{ij}(x_1, \dots, x_n)\frac{\partial^2 f}{\partial x_i\partial x_j} + \sum_i u_i(x_1, \dots, x_n)\frac{\partial f}{\partial x_i}, $$ where $C_{ij}$'s and $u_i$'s are $C^\infty$-functions of $x_1,\dots x_n$, and moreover $C_{ij}(x_1, \dots, x_n)$ is a symmetric positive semi-definite matrix for any $(x_1, \dots, x_n)\in\mathbb R^n$. Suppose also that $C_{ij}$'s and $u_i$'s behave nicely at infinity - if needed, we can assume that they have compact supports, but some other growth condition might be more reasonable. Is the Cauchy problem for such a PDE well-posed? What is a suitable reference for this type of PDEs?
If it simplifies the discussion, I am actually more interested in the case when $M$ is a compact manifold, $f$ a function on $[0,\infty)\times M$, and the PDE is $\frac\partial{\partial t} f = Df$, with $D$ a 2nd-order differential operator on $M$ with positive semi-definite principal symbol and such that $D1=0$ (i.e. in local coordinates it is as the RHS of the PDE written above), when we don't have to care about the behavior at infinity.