The properties of the solutions for Cauchy problem of nonlinear parabolic equations in non-divergent form with density

We investigate the solutions for the following nonlinear degenerate parabolic equation in non-divergent form with density
$$ \left|x\right|^{n} \frac{\partial u}{\partial t} =u^{m} div\left(\left|\nabla u\right|^{p-2} \nabla u\right). $$
We discuss the properties, which are different from those for the equations in divergence form, thus generalizing various known results. Then getting a self-similar solution we show the asymptotic behavior of solutions at $t \to \infty$. Slow and fast diffusion cases are investigated. Finally, we present the results of some numerical experiments.