Solution to non-linear PDE I think I have found a solution for a PDE of the form


*

*$u_t + g(u)u_x = 0$

*where $u(x, 0) = g^{-1}(x)$
The solution is  $u(x,y) = g^{-1}\left(\frac x{t+1}\right)$
This solution satisfies 1 and 2 under the assumption that $\forall z, g\left(g(z)^{-1}\right) = z$
However I am worried about the effects of discontinuities in $g$ or its inverse, and issues where the function is not 1-1. 
What sort of problems should I watch out for and how can I get around them?
 A: This equation you are dealing with, although it looks simple, it is a very problematic one: it is a nonlinear conservation law. These equation DO NOT have smooth solutions for all $t>0$. Their solution break down after a short time and the develop "shock waves" - discontinuous weak solutions.
The simpler such equation is Burger's Equation ($g(u)=u$)
$$
u_t+uu_x=0,
$$ 
and there is a written about it as it is the simplest model for shock formation.
A: The classical solution $$
u(x,t) = g^{-1}\left(\frac{x}{t+1}\right)
$$
follows from the method of characteristics, which requires that the solutions remains smooth along the characteristic curve as time increases. Smoothness is lost at the breaking time $$
t_b = \inf \frac{-1}{(g\circ g^{-1})'(x)} = -1 \, .
$$
This can be seen in the solution's denominator which vanishes at $t=-1$. Therefore, if only positive times are considered, no shock wave will form!
Now, if there are some discontinuities in $g$ or $g^{-1}$, the problem statement makes little sense as $g$ is no longer invertible over its whole domain of definition.
