Reference request: move the source to the initial condition and vice versa in a PDE problem

I am trying to find references in the literature that connect solutions to two problems that are presented below. They deal with deterministic laws of conservation.

Cauchy nonhomogeneous problem:
$$ (1) hspace {1cm} begin {cases}
u_t + (f_ {1} (u)) _ x = lambda cdot g (u) \[2ex]
u (x, 0) = h_ {1} (x)
end {cases}

Homogeneous Cauchy problem:
$$ (2) hspace {1cm} begin {cases}
u_t + (f_ {2} (u)) _ x = 0 \[2ex]
u (x, 0) = h_ {2} (x)
end {cases}

Here u $ in mathbb {R} ^ n $ Y $ lambda $ is a constant If we have initial condition given with:
$$ u (x, 0) = begin {cases}
u_l, x<0 \[2ex] u_r, x>0
end {cases} $$

we would talk about the homogeneous Riemann problem and the non-homogeneous Riemann problem.

I work on a problem in which it would be very useful to switch between problems. $ (1) $ Y $ (2) $ occasionally. So at one point I would like to work on a problem that has a source term $ lambda cdot g (u) $that is, problem $ (1) $. In the other moment I would like to use the coordinate change (or something similar) to be able to work with a problem without a source term and with a different initial condition (and possibly also a different flow), that is, a problem $ (2) $.

If someone knows some paper / book / notes that change problems $ (1) rightarrow (2) $ or $ (2) rightarrow (1) $ or have an idea of ​​how I could do this, please write it down. It would help me a lot.

Also one of the teachers I know told me that he had seen something like that in some articles (but unfortunately it was a long time ago and I could not remember it now). He suggested that I try to use the Duhamels principle (or find some variation that relates to conservation laws).