Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs:


A$\dfrac{\partial}{\partial t}\pmb{v}(t,x)=B(t,x,\pmb{v}(t,x))\dfrac{\partial}{\partial x}\pmb{v}(t,x)+N(x,t)$,


with


$\pmb{v}\_i(t\_0,x\_0)=\pmb{v}\_{i0}$, $\forall (t\_0,x\_0)\in\Gamma\_i$, $i=1,...,n$


where $\pmb{v}$ is the vector of the variables, $A,\;B$ and $N$ are three matrices, the $\pmb{v}\_{i0}$ are continuous in initial curves $\Gamma\_i$, the initial curves $\Gamma\_i$ are non-characteristic and the coefficients of $A$ are constants. In particular, the equations of the system in what I'm really interested have the form:


$0=-\dfrac{\partial b}{\partial x}(t,x)+f\_1(x,b(t,x),c(t,x),d(t,x))$,


$0=-\dfrac{\partial c}{\partial x}(t,x)+f\_2(x,b(t,x),c(t,x),d(t,x))$


$\dfrac{\partial d}{\partial t}(t,x)=-b(t,x)\dfrac{\partial d}{\partial x}(t,x)+f\_3(x,b(t,x),c(t,x),d(t,x))$.


I have only found results for systems in which the matrix $A$ is invertible, but in the case I have, that condition isn't satisfied. So, any suggestion will be really appreciated.



Existence and uniqueness of solutions for a system of first order PDEs
(This is mostly an extended comment to answer [this comment](https://mathoverflow.net/questions/238041/existence-and-uniqueness-of-solutions-for-a-system-of-first-order-pdes?noredirect=1#comment589440_238041).)


Yes, you are right to worry about the curves. If you initial data curve is real analytic, then you can find a local, real analytic change of variables, so that the proof using CK goes through identically. 


When your initial data curve is only smooth, but not real analytic, you will then need to construct a sequence of approximate change of variables and show that the appropriate limit converges. The sequence of approximate change of variables will be based on first approximating your data curve using something like [Weierstrass approximation](https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem) by a sequence of real analytic curves, and then straighten out these approximate curves using a real analytic change of variables. The fact that hyperbolic equations have finite speeds of propagation means that the inevitable spatial localization in this procedure is relatively harmless. 


Unfortunately I am not aware of any place where this argument is written down in full detail. It is one of those folk theorems that "everyone knows is true", and many subparts or special cases of it can be found in many places. 


