# 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).