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 $$ \rho (x)\frac{{\partial u}} {{\partial t}} = div(u^{m - 1} \left| {Du} \right|^{\lambda - 1} Du) + \rho (x)u^p $$ 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.

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