I gave an answer to your earlier question, but although it was technically correct, I see now that since you have only two independent variables my answer wasn't really the right one. Let me try one more time (but I can't guarantee I'm right).

I'm going to address only case 4, since that's the one you care about. Given your assumptions, there exist invertible-matrix-valued functions $P(x,u)$ and $Q(x, u)$ such that the system $$ P(A\partial_1 u + B\partial_2 u) = Pf $$ can be written as $$ \partial_1 v + C(x,v)\partial_2v = g(x,v), $$ where $v = (v_1, v_2, v_3, v_4)$. $$ C(x,v) = \begin{bmatrix}a(x,b)&0&0&0\\ 0&b(x,v)&0&0\\ 0&0&0&-1\\ 0&0&1&0\end{bmatrix} $$ If you fix a rectangular domain, say, $D = [0,T]\times[0,S]$, then this system is well-behaved from the point of view of uniqueness and, with the right additional assumptions, existence if you specify initial data consisting of $v_1$ and $v_2$ on $\{0\} \times [0,S]$ and $v_3$ only on the boundary of $D$ (i.e., {0,T}\times [0,S] \cup [0,T] \times {0,S}$).

This is a mixed hyperbolic-elliptic system that is hyperbolic in $v_1$ and $v_2$ and elliptic in $v_3$ and $v_4$. My guess is that linear systems like this been analyzed and solved before, but unfortunately I don't know any specific reference. You would handle the quasilinear version using the linear theory using the usual techniques (inverse function theorem, fixed point theory, etc.). If you know the theory of linear first order elliptic systems of PDE's and of linear first order hyperbolic systems of PDE's in two independent variables really well, it's reasonably straightforward to adapt it to a coupled system like this.

I gave an answer to your earlier question, but although it was technically correct, I see now that since you have only two independent variables my answer wasn't really the right one. Let me try one more time (but I can't guarantee I'm right).

I'm going to address only case 4, since that's the one you care about. Given your assumptions, there exist invertible-matrix-valued functions $P(x,u)$ and $Q(x, u)$ such that the system $$ P(A\partial_1 u + B\partial_2 u) = Pf $$ can be written as $$ \partial_1 v + C(x,v)\partial_2v = g(x,v), $$ where $v = (v_1, v_2, v_3, v_4) = Q(x,u)u$. $$ C(x,v) = \begin{bmatrix} a(x,b)& 0 & 0 & 0 \newline 0 & b(x,v) & 0 & 0 \newline 0 & 0 & 0 & -1 \newline 0 & 0 & 1 & 0 \end{bmatrix} $$ If you fix a rectangular domain, say, $D = [0,T]\times[0,S]$, then this system is well-behaved from the point of view of uniqueness and, with the right additional assumptions, existence if you specify initial data consisting of $v_1$ and $v_2$ on $\{0\} \times [0,S]$ and $v_3$ on the boundary of $D$.

This is a coupled hyperbolic-elliptic system that is hyperbolic in $v_1$ and $v_2$ and elliptic in $v_3$ and $v_4$. My guess is that linear systems like this been analyzed and solved before, but unfortunately I don't know any specific reference. You would handle the quasilinear version using the linear theory using the usual techniques (inverse function theorem, fixed point theory, etc.). If you know the theory of linear first order elliptic systems of PDE's and of linear first order hyperbolic systems of PDE's in two independent variables really well, it's reasonably straightforward to adapt it to a coupled system like this.

I gave an answer to your earlier question, but although it was technically correct, I see now that since you have only two independent variables my answer wasn't really the right one. Let me try one more time (but I can't guarantee I'm right).

I'm going to address only case 4, since that's the one you care about. Given your assumptions, there exist invertible-matrix-valued functions $P(x,u)$ and $Q(x, u)$ such that the system $$ P(A\partial_1 u + B\partial_2 u) = Pf $$ can be written as $$ \partial_1 v + C(x,v)\partial_2v = g(x,v), $$ where $v = (v_1, v_2, v_3, v_4)$. $$ C(x,v) = \begin{bmatrix}a(x,b)&0&0&0\\ 0&b(x,v)&0&0\\ 0&0&0&-1\\0&0&1&0\end{bmatrix} $$ If you fix a rectangular domain, say, $D = [0,T]\times[0,S]$, then this system is well-behaved from the point of view of uniqueness and, with the right additional assumptions, existence if you specify initial data consisting of $v_1$ and $v_2$ on $\{0\} \times [0,S]$ and $v_3$ only on the boundary of $D$ (i.e., {0,T}\times [0,S] \cup [0,T] \times {0,S}$).

This is a mixed hyperbolic-elliptic system that is hyperbolic in $v_1$ and $v_2$ and elliptic in $v_3$ and $v_4$. My guess is that linear systems like this been analyzed and solved before, but unfortunately I don't know any specific reference. You would handle the quasilinear version using the linear theory using the usual techniques (inverse function theorem, fixed point theory, etc.). If you know the theory of linear first order elliptic systems of PDE's and of linear first order hyperbolic systems of PDE's in two independent variables really well, it's reasonably straightforward to adapt it to a coupled system like this.

I gave an answer to your earlier question, but although it was technically correct, I see now that since you have only two independent variables my answer wasn't really the right one. Let me try one more time (but I can't guarantee I'm right).

I'm going to address only case 4, since that's the one you care about. Given your assumptions, there exist invertible-matrix-valued functions $P(x,u)$ and $Q(x, u)$ such that the system $$ P(A\partial_1 u + B\partial_2 u) = Pf $$ can be written as $$ \partial_1 v + C(x,v)\partial_2v = g(x,v), $$ where $v = (v_1, v_2, v_3, v_4)$. $$ C(x,v) = \begin{bmatrix}a(x,b)&0&0&0\\ 0&b(x,v)&0&0\\ 0&0&0&-1\\ 0&0&1&0\end{bmatrix} $$ If you fix a rectangular domain, say, $D = [0,T]\times[0,S]$, then this system is well-behaved from the point of view of uniqueness and, with the right additional assumptions, existence if you specify initial data consisting of $v_1$ and $v_2$ on $\{0\} \times [0,S]$ and $v_3$ only on the boundary of $D$ (i.e., {0,T}\times [0,S] \cup [0,T] \times {0,S}$).

This is a mixed hyperbolic-elliptic system that is hyperbolic in $v_1$ and $v_2$ and elliptic in $v_3$ and $v_4$. My guess is that linear systems like this been analyzed and solved before, but unfortunately I don't know any specific reference. You would handle the quasilinear version using the linear theory using the usual techniques (inverse function theorem, fixed point theory, etc.). If you know the theory of linear first order elliptic systems of PDE's and of linear first order hyperbolic systems of PDE's in two independent variables really well, it's reasonably straightforward to adapt it to a coupled system like this.