I am trying to verify the well-posedness of some PDEs which I solve numerically. There is a paper which proves the well-posedness of a similar problem using Ladyzhenskaya's book on hyperbolic problems. In particular, it uses Section 6, Chapter V to show that an equation of the form
$\frac{\partial \rho}{\partial t} - \nabla \cdot \mathbf{a}\left( \mathbf{x},\rho,\nabla \rho \right) + a\left(\mathbf{x},t,\nabla \rho,\rho \right) = 0$,
is well-posed.
If that problem is well-posed, is it also true that
$- \nabla \cdot \mathbf{a}\left( \mathbf{x},\rho,\nabla \rho \right) + a\left(\mathbf{x},\nabla \rho,\rho \right) = 0$
is well-posed?
Can you please give me either a proof or where to look for the result. I suspect that there is such a result in Ladyzhenskaya's book on elliptic problems, but I am away from my university, so I don't have direct access to it right now. I appreciate any help you can provide in finding this result.