ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS OF THE INHOMOGENEOUS POROUS MEDIUM EQUATION WITH CRITICAL VANISHING DENSITY.

We study the long-time behavior of non-negative, finite-energy solutions to the initial value problem for the Porous Medium Equation with variable density, i.e. solutions of the problem {ρ(x)^αt^u = Δ u^m, in Q := ℝⁿ x ℝ₊, u(x,0) = u₀(x), in ℝⁿ, where m > 1, u₀ ∈ L¹ (ℝⁿ, ρ (x)dx) and n ≥ 3. We assume that ρ (x) ~ C|x|^-2 as |x| → ∞ in ℝn. Such a decay rate turns out to be critical. We show that the limit behavior can be described in terms of a family of source-type solutions of the associated singular equation |x|^-2 ut = Δu^m . The latter have a self-similar structure and exhibit a logarithmic singularity at the origin.Δ [ABSTRACT FROM AUTHOR]