Second-order asymptotics for the fast-diffusion equation
Robert J. McCann, Dejan Slepčev · International Mathematics Research Notices · 2006
In many dissipative settings, initial disturbances will gradually disappear and all but their crudest features—such as size and location—will eventually be forgotten. Quantifying the rate at which this information is lost is sometimes a question of central interest. Here this question is addressed for the fast-diffusion equation ∂v/∂τ = Δ(v1−2/(n+p)), τ, v(y, τ) ≥ 0, y ∈ ℛn, in the range p ≥ n ≥ 2 ≠ p of nonlinearities. We use displacement convexity and dissipation of entropy to show any two solutions starting with finite pth moments and the same total mass converge at rate ‖v(·,τ)−v˜(·,τ)‖L1(ℛn)=(a/τα)|z0−z˜0|+O(1/τ1−δ) as τ → ∞, for any δ > 0. Here α = (1/2)(1 + n/p), a depends only on n, p, and ‖v0‖L1(ℛn)′, while z0 denotes the center of mass of v0(·) = v(·, 0). In contrast, for |p| < n, we show the entropy is not displacement semiconvex, even near equilibrium.