In the study of nonlinear PDEs, I often work on conservation laws and a lot of time I work on the two problems given bellow:

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

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

Here u $\in \mathbb{R}^n$ and $\lambda$ is a constant. Besides $u$, the functions $g,f_1, f_2$ could also depend of $x$ and/or $t$.

A long time ago one professor of mine told me that he read somewhere that

**there is a way of transforming problem $(1)$ to the problem $(2)$ for some special cases of functions $g,f_1, f_2$. And vice versa.**

But he can't remember where he has read that.

**It would be very useful if we can switch between problems $(1)$ and $(2)$ from time to time.**

So in one moment you work on a problem that has a source term $\lambda \cdot g(u)$, i.e. problem $(1)$. In the other moment you use change of coordinates (or something similar) and now you work with a problem with no source term and with different initial condition (and possibly different flux too), i.e. problem $(2)$.

**Of course I am pretty sure that some general transformation doesn't exist.**

In the problem $(1)$ source is time dependent and in the problem $(2)$ new initial data are not time dependent. But

- maybe source could go partially in the flux and partially into the initial data or something similar.
- Or maybe this kind of transformation exists for some functions $g,f_1, f_2, h_1, h_2$ with the special properties.
- Or maybe it exists for some other kind of PDE problems - that don't deal with conservation laws.

**The only example of that transformation that I know would be this:**

if $g(u)=(w(u))_x$ (assuming that function $w$ could be found), then the source term would be absorbed in the flux term. So we would get the problem $(2)$ form where $f_2 (u) = f_1 (u) - \lambda \cdot w(u)$ and $h_2(x)=h_1 (x).$

So if anyone knows any reference in the literature that deals with this issue share it. Of course you could share your ideas or answers too. This question intrigues me from the moment the professor told me that.