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

We study the long-time behavior of nonnegative solutions to the Cauchy problem {ρ(x) ∂^tu = Δ_u^m in Q := ℝ^N x R₊ u(x, 0) = u_o in ℝ^N 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)u_o dx < ∞. We show that in this case nontrivial solutions to the problem have a longtime universal behavior in separate variables of the form u(x,t)∼i^∼1/(m-1)W(x), where V = W^m is the unique bounded, positive solution of the sublinear elliptic equation-ΔV = cp(x)V¹/^m in ℝ^N vanishing as |x| → ∞; c = l/(m - 1). Such a behavior of u is typical of Dirichlet problems on bounded domains with zero boundary data. It strongly departs from the behavior in the case of slowly decaying densities, ρ(x) ∼ |x|^-γ as |x| → ∞ with 0 ≤ γ ≤ 2, previously studied by the authors. If ρ(x) has an intermediate decay, ρ ∼ |x|^-γ as |x| → ∞ with 2 < γ < γ2 := N - (N - 2)/m, solutions still enjoy the finite propagation property (as in the case of lower γ). In this range a more precise description may be given at the diffusive scale in terms of source-type solutions U(x, t) of the related singular equation |x| ^-γu_t = Δu^m. Thus in this range we have two different space-time scales in which the behavior of solutions is non-trivial. The corresponding results complement each other and agree in the intermediate region where both apply, thus providing an example of matched asymptotics.