On the behavior of solutions to the Cauchy problem for a degenerate parabolic equation with inhomogeneous density and a source

The Cauchy problem for a degenerate parabolic equation with a source and inhomogeneous density of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \rho (x)\frac{{\partial u}} {{\partial t}} = div(u^{m - 1} \left| {Du} \right|^{\lambda - 1} Du) + \rho (x)u^p $$\end{document} is studied. Time global existence and nonexistence conditions are found for a solution to the Cauchy problem. Exact estimates of the solution are obtained in the case of global solvability.