I've made a quick search but I haven't been able to find anything regarding a particular type of this (first order) linear, non-strictly hyperbolic system of PDEs.
$$u_t+A(t)u_x=h(t,u,x)$$
where $A(t)$ is an scaled version of the $(2\times 2)$-identity matrix, this is both eigenvalues are $f(t)$.
In general literature conditions are given for the existence of the solution of a strictly hyperbolic system, but clearly those do not apply here.
Do you know if it is possible to show the existence of a solution for this system? (under certain conditions for $f$ and $h$ of course).
This may be some well-known result, but this is not my field of study and I am rather ignorant, so please let me know if the question is trivial or non well-posed.
Thanks in advance
Let's think in terms of characteristics, and assume a parametrization $(x(t), t)$ of the coordinates such that $x'(t) = f(t)$. Using the chain rule for the time-derivative of $u(x(t), t)$, we find $$ \frac{\text d}{\text d t} u = u_t + f(t) u_x = h \, . $$ Local existence and uniqueness results follow from the study of the ODE system $$ \begin{aligned} x'(t) &= f(t)\\ u'(t) &= h\big(t, u(t), x(t)\big) \end{aligned} $$ Here, we have used the fact that $A = f\, I_2$ is proportional to the identity matrix, as noted by @Calvin Khor in the comments section. It is one particular example where the method of characteristics applies to PDE systems.