The Cauchy problem for a degenerate parabolic equation with inhomogeneous density and source in the class of slowly decaying initial data

Given a degenerate parabolic equation of the form $\rho(x) u_t=\operatorname{div}(u^{m-1}|Du|^{\lambda-1}Du)+\rho(x)u^p$ with a source and inhomogeneous density, we consider the Cauchy problem with an initial function slowly tending to zero as $|x| \to \infty$. We find conditions for the global-in-time existence or non-existence of solutions of this problem. These conditions depend essentially on the behaviour of the initial data as $|x|\to \infty$. In the case of global solubility we obtain a sharp estimate of the solution for large values of time.