I am still pretty new to the field of PDEs, but I wanted to develop a novel economics model to study the spatial congestion of economic activity. For example, we see that cities tend to accumulate more and more economic activity, leading to higher incomes and rental prices, etc. There are also diffusive factors, as things like traffic or living costs push people out of the city.
I was trying to find a PDE model that has terms to model both the congestive and diffusive forces. The diffusion piece is easy since it is just the standard second derivative in space--so that is not an issue. But the congestive term I am not sure about. The closest model I can find is a backward parabolic equation, or a backward diffusion term, such as $u_t = - \Delta u$. However, according to Wikipedia, backward parabolic terms are not well-posed, and will likely lead to convergence problems with the numerical solver. Right now, the model in 1D would look something like:
$$ u_t = d(x)u_{xx} - c(x)u_{xx} $$
This is still a very simplified model. I have not developed the terms for the $c(x), d(x)$ functions yet, but I imagine they will be anisotropic. Are there any better models that I can use for this congestive or agglomerative process that are more stable, and can be computed with standard PDE solvers, like FEM?