Large-time behaviour of the higher-dimensional logarithmic diffusion equation

Abstract

Let n ≥ 3 and let ψ_λ0 be the radially symmetric solution of Δ log ψ + 2βψ + βx · ∇ψ = 0 in ℝⁿ, ψ(0) = λ(0), for some constants λ₀ > 0, β > 0. Suppose u₀ ≥ 0 satisfies u₀ − ψ_λ0 ∈ L¹ (ℝⁿ) and u₀ (x) ≈ (2(n − 2)/β)(log∣x∣/∣x∣²) as ∣x∣ → ∞. We prove that the rescaled solution ũ(x,t) = e^2βtu(e^βtx, t) of the maximal global solution u of the equation u_t = Δ log u in ℝⁿ × (0, ∞), u(x, 0) = u₀ (x) in ℝⁿ, converges uniformly on every compact subset of ℝⁿ and in L¹ (ℝⁿ) to ψ_λ0 as t → ∞. Moreover, ∥ũ(·, t) − ψ_λ0∥_L¹(ℝⁿ) ≤ e^{−(n−2)βt}∥u₀ − ψ_λ0∥_L¹(ℝⁿ) for all t ≥ 0.