I am searching for a reference for the general (uniformly) parabolic Cauchy problem of second order, that is

\begin{align*} \partial_t u - \sum_{1\leq i,j\leq N}\partial_{x_j}(a^{ij}\partial_{x_i}) + \sum_{1\leq i\leq N} b^i \partial_{x_i}u +cu &=f,\\ u(0,\cdot) = u^0, \end{align*} where the coefficients $a^{ij}, $$b^i$ and data $f$, $u^0$ are smooth (and the usual uniform elliptic estimate for the $a^{ij}$s), but when this system is considered either on the whole space $\mathbb{R}^N$ or the torus $\mathbb{T}^N$ (so : no boundaries conditions).

This is a purely bibliographic question since I am quite sure (I think ...) that the analysis of this Cauchy problem should mimick exactly the initial-boundary value one on $[0,T]\times\Omega$, only dropping off the assumptions regarding the boundary (well, for $\mathbb{R}^N$ I imagine that one should add an assumption on the behavior of the data at infinity).

There is a bench of references for the boundary-value problem, but nothing for the torus and for the whole space the only one I ran into assumes that the coefficients do not depend on the time variable (Krylov).

Any simple references for this rather academic setting ?

Thanks.

falseon unbounded domains. For instance, there are nontrivial smooth solutions to the initial-value problem for the classical heat equation on $\mathbb{R}^n$ with initial values zero. To get uniqueness, you have to impose conditions on thesolution(e.g. growth, positivity, integrability, etc), not just on the initial data. $\endgroup$