On the behavior of solutions of the Cauchy problem for a degenerate parabolic equation with source in the case where the initial function slowly vanishes

We study the Cauchy problem for a degenerate parabolic equation with source and inhomogeneous density of the form in the case where the initial function slowly vanishes as | x | → ∞: We establish conditions for the existence and nonexistence of a global (in time) solution. These conditions strongly depend on the behavior of the initial data as | x | → ∞: In the case of global solvability, we establish a sharp estimate of the solution for large times.