Suppose we are given a pde: $\frac{\partial u}{\partial t} +\frac{\partial {(b(x)u)}}{\partial x}=0$. Let $u(t,x)\in C^1(\mathbb{R}^2)$ be a solution and $x=X_1(t)$ and $x=X_2(t)$ be two characteristic curves such that $X_1(t)<X_2(t)$.
I have shown that $\int_{X_1(t)}^{X_2(t)}u(t,x)dx$ is constant.
My question is: what can we say about the solution $u(t,x)$?(i mean using that the integral is constant)
Any hints are welcome!
Thank you!