I'm looking here for a reference to solve the following equation with the method of characteristic $$ \partial_t U + A\partial_x U = f$$ $$U(t=0,x)=U_0(x)$$ $$B(t,x=0)U = 0$$ where $(t,x)\in [0,T]\times \mathbb{R}^+$, $A$ is $n\times n$ matrix with constant coefficient with $\det A\neq 0$ with positive eigenvalues and $B$ a $n\times n$ matrix with constant coefficient with $\det B \neq 0$. $f\in L^2((0,T)\times \mathbb{R}^+)$, $U_0\in L^2(\mathbb{R}^+)$. Everywhere I look the characteristic method is developed for regular solutions.
Can anyone provide a clear reference for the method of characteristic with data in $L^2$ on this example (if possible) or for another example (where it gives the existence and the uniqueness of a solution)?