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$$u_t = \text{div}(ρ(x)u^{m-1}|Du|^{λ-1}Du) + u ^p $$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.