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 - Martynenko, A. V.; Tedeev, A. F.; Shramenko, V. N. (Ukrains’kyi Matematychnyi Zhurnal, 2012)
We study the Cauchy problem for a degenerate parabolic equation with source and inhomogeneous density of the form u_t = div(ρ(x)u^m-1|Du|^λ-1Du) + u ^p in the case where initial function decreases slowly to zero as |x| → ∞. We establish conditions for the existence and nonexistence of a global-in-time solution, which substantially depend on the behavior of the initial data as |x| → ∞. In the case of global solvability, we obtain an exact estimate of a solution for large times.