Consider the flat torus $\mathbf{T}^d:=\mathbf{R}^d/\mathbf{Z}^d$ and define the corresponding periodic-parabolic cylinder $Q_T:=(0,T)\times\mathbf{T}^d$.
Given a matrix field $A:Q_T\rightarrow\text{M}_N(\mathbf{R})$, I am interested in the Cauchy problem for the following system of equations
\begin{align*} \partial_t U -\sum_{k=1}^d \partial_k(A\,\partial_k U) =F, \end{align*} where $F:Q_T\rightarrow\mathbf{R}^N$ is given together with some initial data $U^0$. The unknown $U$ is a vector field over $Q_T$ with values in $\mathbf{R}^N$. If my understanding of the literature is correct, there's an ellipticity condition, for parabolic systems, attributed to Petrovskii and which in the precise case takes the following form.
Definition : The matrix field $A$ is said to satisfy uniformly Petrovskii's ellipticity condition on $Q_T$ if there exists $\delta>0$ such that for all $(t,x)\in Q_T$, the spectrum of $A(t,x)$ lies in the $\{z\in\mathbf{C}\,:\,\text{Re}(z)\geq \delta\}$.
This ellipticity condition does not ensure an energy estimate for the solution $U$: if $A(t,x)$ is not symmetric, it could happen that it is non-positive. However, in the specific case of a constant matrix field, one checks by elementary Fourier analysis (in the space variable) that this condition is sufficient to build a solution for generic data $F$ and $U^0$.
So far I found the following references about parabolic systems:
[1] Linear and Quasi-linear Equations of Parabolic Type (Ladyzenskaja, Solonnikov, Uralceva)
[2] Parabolic systems (Eidel'man)
[3] Partial differential equations of parabolic type (Friedman)
[4] Dynamic theory of quasilinear parabolic systems (Amann)
Unfortunately [2] does not cover the divergence case (when $A$ is only continuous, say). For [1] and [3] the strategy is to compute a fundamental solution and establish sufficient decay estimate. I find this approach quite involved. Finally [4] uses some semi-group theory to exploit the Petrovskii condition but again, the scope is far more general and I would be happy to find another approach.
I am looking for other references about parabolic systems in divergence form, which uses this Petrovskii condition to recover well-posedness of the system. I would be particulary interested in a perturbative approach using a freezing argument.
