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

Inhomogeneous Cauchy 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 $\in \mathbb{R}^n$ and $\lambda$ 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 homogeneous Riemann problem and inhomogeneous Riemann problem.

I work on a problem in which it would be very useful to switch between problems $(1)$ and $(2)$ from time to time. So in one moment I would like to work on a problem that has a source term $\lambda \cdot g(u)$, i.e. problem $(1)$. In the other moment I would like to use change of coordinates (or something similar) so I can work with a problem with no source term and with different initial condition (and possibly different flux too), i.e. problem $(2)$.

**If anyone knows any paper/book/notes that change from problems $(1)\rightarrow (2)$ or $(2)\rightarrow (1)$ or has an idea how could I do this, please write it. It would help me a lot.**

Also one of the professor I know have said to me that he have seen something like that in a few papers (but unfortunatelly it was a long time ago and he couldn't remember it now). He suggested me to try to use Duhamels's principle (or to find some variation of it that deals with conservation laws).