Document Type
Lecture
Publication Date
4-5-2012
Abstract
We study the intermediate asymptotics of non-negative solutions to the Cauchy problem (P): ρ(x)·∂ₜu = Δuᵐ in Q = ℝⁿ × ℝ⁺, with u(x, 0) = u₀ in dimensions n ≥ 3. Assume m > 1 (slow diffusion), and ρ(x) is positive, bounded, and behaves like |x|⁻ᵞ as |x| → ∞, with γ > 0. The initial data u₀ is non-negative and satisfies ∫ ℝⁿ ρ(x)u₀ dx < ∞. The asymptotic behavior depends on whether 0 < γ < 2 or γ > 2. For 0 < γ < 2, solutions resemble source-type solutions U_E(x, t) = t^–α·F_E(xt^–β) for a related singular problem. For γ > 2, the solution behaves as u(x, t) ~ t^–¹⁄(m–1)·W(x), where V = Wᵐ solves –ΔV = cρ(x)V¹⁄ᵐ with c = 1⁄(m–1). For intermediate decay 2 < γ < γ₂ := N – (N–2)/m, finite propagation still holds. The matching asymptotics describe behavior across different space-time scales. (References: Reyes & Vázquez (2009); Kamin, Reyes & Vázquez (2010))
Relational Format
presentation
Recommended Citation
Reyes, Guillermo, "The Cauchy Problem for The Porous Medium Equation with Variable Density. Asymptotic Behavior of Solutions" (2012). Colloquium. 36.
https://egrove.olemiss.edu/math_colloquium/36
