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

$\pmb{v}(t_0,x_0)=\pmb{v}_0$,

where $\pmb{v}$ is the vector of the variables, $A,\;B$ and $N$ are three matrices, 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.

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.

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

$\pmb{v}(t_0,x_0)=\pmb{v}_0$,

where $\pmb{v}$ is the vector of the variables, $A,\;B$ and $N$ are three matrices, 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.

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

$\pmb{v}(t_0,x_0)=\pmb{v}_0$,

where $\pmb{v}$ is the vector of the variables, $A,\;B$ and $N$ are three matrices, 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.