I'm a physicist and during my research work, I found a system of linear pde with non constant coefficients that I have to study, since I have totally no experience about systems of pde and I have even problems to find some references. I would be really great to have some advice about how to face the problem. What I would like to do is of course find a solution of this system but even proving the existence of the solution can be a first great step. So thank you in advance for any possible help and some good reference. Here in the link you can find the system:
[System of pde](https://i.stack.imgur.com/tc85P.jpg)


$$\begin{pmatrix} 0 & f\_1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & f\_1 \end{pmatrix}\partial\_x \begin{pmatrix} N\_1 \\ N\_2 \\ N\_3 \end{pmatrix} + \begin{pmatrix} f\_2 & 0 & 0 \\ 0 & 0 & f\_2 \\ 0 & 0 & 0 \end{pmatrix}\partial\_y \begin{pmatrix} N\_1 \\ N\_2 \\ N\_3 \end{pmatrix} + \begin{pmatrix} 0 & 0 & 0 \\ 0 & f\_3 & 0 \\ f\_3 & 0 & 0 \end{pmatrix}\partial\_z \begin{pmatrix} N\_1 \\ N\_2 \\ N\_3 \end{pmatrix}= \begin{pmatrix} \omega\_1 \\ \omega\_2 \\ \omega\_3\end{pmatrix}$$


Where we have three independent variables $(x,y,z)$ and three dependent variables $(N\_1,N\_2,N\_3)$. The functions $\omega\_i:R^3\rightarrow R$ $f\_i: R^3\rightarrow R $, $N\_i:R^3\rightarrow R $ are all smooth functions. Given that the matrix of coefficients are all singular I don't know how to proceed even about the classification of this system. Like I said I would be interested in a local solution or at least the proof of the existence of the solution. Any reference or help about this system will be really appreciated.


\*Edit, sorry but for some reason the editor was not allowing me to write the matrices that I need in a good way, so I put a link to the image of the system that I create with latex.



System of linear pde with non constant coefficients
Recall that a first order system $A\_{ij}^\mu \partial\_\mu \phi^j + B\_{ij} \phi^j = 0$ (the right hand-side could also be inhomogeneous) is *symmetric hyperbolic* when there exists at least one covector $p\_\mu$ such that $A\_{ij}^\mu p\_\mu > 0$, the contraction is a positive definite symmetric matrix. If the coefficients $A\_{ij}^\mu = A\_{ij}^\mu(x)$ are $x$-dependent, then the positivity condition should be satisfied for all $x$. There are no special conditions imposed on $B\_{ij}$, which could also be $x$-dependent. This is all standard. And the initial value problem for such systems is well posed, at least on those domains where the causal structure of formed by the cones of $p\_\mu$'s satisfying the above positivity conditions is globally hyperbolic (the domain admits a foliation by Cauchy surfaces). See for instance the excellent account in Chapter 7 of [1].


Let me now introduce some non-standard terminology. I call a linear system $$A\_{ij}^{\mu\_1\cdots \mu\_k} \partial\_{\mu\_1} \cdots \partial\_{\mu\_k} \phi^j + B(\phi,\partial\phi, \cdots \partial^{k-1}\phi) = 0 \tag{\*}$$ *higher order symmetric hyperbolic* when it can be put in symmetric hyperbolic form by reduction to first order. Next, I call a linear system (using the notation $A\_{ij}(\partial) = A\_{ij}^{\mu\_1\cdots \mu\_k} \partial\_{\mu\_1} \cdots \partial\_{\mu\_k}$ and writing l.o.t for what I wrote as $B\_{ij}(\cdots)$ above) $$A\_{ij}(\partial) \phi^j + l.o.t = 0$$ *generalized symmetric hyperbolic* when there exists a complementary operator $C\_k^k(\partial) = (C\_k^i)^{\mu\_1\cdots \mu\_l} \partial\_{\mu\_1} \cdots \partial\_{\mu\_l}$ such that applying it to (\*) gives a system $$ C\_k^i(\partial) A\_{ij}(\partial) \phi^j + l.o.t = 0$$ that is higher order symmetric hyperbolic.


The point is that a generalized symmetric hyperbolic system has an equally well-posed initial value problem, with or without a source term, as a symmetric hyperbolic one. Unfortunately, I don't have a reference for this, except for the parallel discussion that I gave for *generalized normally hyperbolic* systems in the recent paper [2] (just replace "normally hyperbolic" by "symmetric hyperbolic" everywhere).



> 
> **Claim:** Your PDE system is generalized symmetric hyperbolic.
> 
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> 


Let me first illustrate by a relevant example, what it means for a system to be higher order symmetric hyperbolic. Take the equation $$\delta\_{ij} \partial\_x \partial\_y \partial\_z N^j + l.o.t = 0 , \tag{\*\*}$$
where $\delta\_{ij}$ is just the Kronecker delta. By introducing the auxiliary variables $N\_z^j = \partial\_z N^j$ and $N\_{yz}^j = \partial\_y N\_z^j$, it can be reduced to the first order system
$$\begin{pmatrix}
 \delta\_{ij}\partial\_z & 0 & 0 \\
 0 & \delta\_{ij}\partial\_y & 0 \\
 0 & 0 & \delta\_{ij}\partial\_x
\end{pmatrix}
\begin{pmatrix} N^j \\ N\_z^j \\ N\_{yz}^j \end{pmatrix} + l.o.t = 0 ,$$ where the desired positivity is satisfied by any covector from the first octant, $p\_x,p\_y,p\_z>0$. This shows that (\*\*) is indeed higher order symmetric hyperbolic.


Your PDE system has the special form $$\begin{pmatrix}
 f\_2 \partial\_y & f\_1 \partial\_x & 0 \\
 0 & f\_3 \partial\_z & f\_2 \partial\_y \\
 f\_3 \partial\_z & 0 & f\_1 \partial\_x
\end{pmatrix}
\begin{pmatrix}
 N\_1 \\ N\_2 \\ N\_3
\end{pmatrix}
= \begin{pmatrix}
 \omega\_1 \\ \omega\_2 \\ \omega\_3
\end{pmatrix} .$$
As mentioned in Deane Yang's answer, this equation cannot be put into symmetric hyperbolic form. That is, there exists no matrix $C\_k^i$ such that the principal symbol $C\_k^i A\_{ij}^\mu p\_\mu$ satisfies both the *symmetry* and *positivity* conditions. You can find $C\_k^i$ such that the symbol becomes symmetric, but not positive. At least one eigenvalue of the resulting $p$-dependent matrix will always remain negative.


However, assuming that $f\_1,f\_2,f\_3 \ne 0$ are nowhere vanishing, one can multiply this system by a second order matrix differential operator (which you can read off from the formula below) such that the system becomes
$$\partial\_x \partial\_y \partial\_z
\begin{pmatrix}
 N\_1 \\ N\_2 \\ N\_3
\end{pmatrix} + l.o.t
= \frac{1}{2 f\_1 f\_2 f\_3} \begin{pmatrix}
 f\_1 f\_3 \partial\_x \partial\_z & -f\_1^2 \partial\_x^2 & f\_1 f\_2 \partial\_x \partial\_y \\
 f\_2 f\_3 \partial\_y \partial\_z & f\_1 f\_2 \partial\_x \partial\_y & -f\_2^2 \partial\_y^2 \\
 -f\_3^2 \partial\_z^2 & f\_1 f\_3 \partial\_x \partial\_z & f\_2 f\_3 \partial\_y \partial\_z
\end{pmatrix}
\begin{pmatrix}
 \omega\_1 \\ \omega\_2 \\ \omega\_3
\end{pmatrix} .$$ From the example (\*\*) it should now be obvious that this system is higher order symmetric hyperbolic, verifying the claim that your system is generalized symmetric hyperbolic. Therefore, it has a well-posed initial-value problem on any hyperplane whose conormal $p\_\mu$ lies in the first octant.



> 
> [1] *Ringström, Hans*, [**The Cauchy problem in general relativity.**](http://dx.doi.org/10.4171/053), ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-053-1/pbk). xiii, 294 p. (2009). [ZBL1169.83003](https://zbmath.org/?q=an:1169.83003).
> 
> 
> [2] See the discussion around Lemma 3 in *García-Parrado, Alfonso and Khavkine, Igor,* [Conformal Killing Initial Data](https://doi.org/10.1063/1.5126683), Journal of Mathematical Physics **60** 122502 (2019). [[arXiv:1905.01231](https://arxiv.org/abs/1905.01231)]
> 
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> 


