This question is cross-posted from Math StackExchange (link). I'm not sure it qualifies as research-level mathematics (although the application is to research) but it has been on MSE for several days, with a bounty for six of them, and failed to garner any sort of response.

It concerns a system of $n$ $n$-dimensional PDEs. \begin{equation*} -\frac{1}{2}\sum_{jk}a_{jk}x_{j}x_{k}u^{i}_{x_{j}x_{k}} +\sum_{jk} u^{j}\pi_{jk}u^{i}_{x_{k}} +c_{i}u^{i}=0. \end{equation*} Each $c_{i}$ is a positive real number. The matrix $A=\left(a_{jk}\right)$ is constant and positive definite while $\Pi=\left(\pi_{jk}\right)$ is constant with $1$s along the diagonal and nonpositive entries elsewhere such that each row sums to zero. Each component of $\mathbf{u}$ is given, with bounded nonnegative value, on the boundary of the domain $U\subset\mathbb{R}_{+}^{n}$.

These equations derive from a model of risk propagation on a network and a system of parabolic PDEs which might in theory be solved forwards in time. \begin{equation*} u^{i}_{t} -\frac{1}{2}\sum_{jk}a_{jk}x_{j}x_{k}u^{i}_{x_{j}x_{k}} +\sum_{jk} u^{j}\pi_{jk} u^{i}_{x_{k}}+c_{i}u^{i}=0. \end{equation*} But I don't have an initial condition and it's the long-time behaviour that I am interested in. That's why I am looking for a time-invariant solution. If there's a name for any of this I'd like to know. Maybe the time-dependent version looks a bit like Burgers'?

I have been reading Evans's book. But it seems there are about a thousand different ways to treat PDEs and I have no idea which is most appropriate to this system. I want to know about existence and uniqueness of solutions, and whether a "comparison" principle holds (i.e. whether $\mathbf{u}\preceq\mathbf{v}$ on $\partial U$ $\implies$ $\mathbf{u}\preceq\mathbf{v}$ in $U$). But any insight would be welcome, even if it depends on some additional assumption, e.g. $\Pi$ symmetric.

"If all you have is a hammer, everything looks like a nail." I am a poor mathematician in general but I do know of the Banach fixed point (contraction mapping) theorem. Could I build a sequence of solutions fixing $\sum_{j} u^{j}\pi_{jk}$ or $\sum_{jk} u^{j}\pi_{jk}u^{i}_{x_{k}}$ from the previous iteration? I guess in order to prove a contraction I would have to quantify the sensitivity of the solution to a change in that term.

A commenter has kindly suggested a change of variables to $\xi_{j}=\log x_{j}$. Then the differential equation becomes $$ -\frac{1}{2}\sum_{jk}a_{jk}u^{i}_{\xi_{j}\xi_{k}} +\sum_{k}\left( \frac{1}{2}a_{kk}+ e^{-\xi_{k}}\sum_{j} u^{j}\pi_{jk} \right)u^{i}_{\xi_{k}}+c_{i}u^{i} =0.$$

each component $u^i$ cannot have a positive interior local max nor a negative interior local min. This doesn't immediately imply a comparison principle since your equation is non-linear. $\endgroup$