Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density

Abstract

We study the long-time behavior of nonnegative solutions to the Cauchy problem {ρ(x) ∂tu = Δum in Q := ℝN x R+ u(x, 0) = uo in ℝN in dimensions N ≥ 3. We assume that m > 1 and ρ(x) is positive and bounded with ρ(x) ≤ Clxl-γ as |x| → ∞ with γ ≥ 2. The initial data uo we nonnegative and have finite energy, i. e., f ρ(x)uo dx < γ