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

Abstract

We study the Cauchy problem for a degenerate parabolic equation with source and inhomogeneous density of the form $$u_t = \text{div}(\rho(x)u^{m-1}|Du|^{\lambda-1}Du) + u ^p $$ in the case where initial function decreases slowly to zero as $|x| \rightarrow \infty$. 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 $|x| \rightarrow \infty$. In the case of global solvability, we obtain an exact estimate of a solution for large times.