Asymptotic expansion of solutions to the drift–diffusion equation with fractional dissipation

Abstract

Abstract. The initial-value problem for the drift-diffusion equation arising from the model of semiconductor device simulations is studied. The dissipation on this equation is given by the fractional Laplacian (−∆) θ/2 . Large-time behavior of solutions to the drift-diffusion equation with 0 < θ ≤ 1 is discussed. When θ > 1, large-time behavior of solutions is known. However, when 0 < θ ≤ 1, the perturbation methods used in the preceding works would not work. Large-time behavior of solutions to the drift-diffusion equation with 0 < θ ≤ 1 is discussed. Particularly, the asymptotic expansion of solutions with high-order is derived.

Asymptotic Behavior of Solutions to the Drift-Diffusion Equation with Critical Dissipation

“…This function is well defined on C((0, ∞); L 1 (R 2 ) ∩ L ∞ (R 2 )) and is not zero (those facts are confirmed by the similar argument as in [25,Proposition 2.9]). Furthermore, the fact that…”

“…When 1 < θ ≤ 2, the drift-diffusion equation with the fractional Laplacian ∂ t u + (−Δ) θ/2 u − ∇ · (u∇ψ) = 0 is treated as a parabolic equation. Indeed, the L p theory for parabolic equations provides well-posedness, global existence, decay and asymptotic expansion as t → ∞ of solutions to the drift-diffusion equation in this case (we refer to [1,2,7,11,[15][16][17]22,25]). When θ = 1, the dissipation balances the nonlinear effect, which is called the critical case.…”

Large-time asymptotics for a matrix spin drift-diffusion model

“…The results were improved in [26], showing optimal L p (R d ) decay estimates. The asymptotic profile to drift-diffusion-Poisson equations with fractional diffusion was analyzed in [32,38].…”

Asymptotic Expansions of Solutions of Fractional Diffusion Equations

“…For the case 0 ≤ K < θ, see [18]. (For related results, see e.g., [20], [22], [26], [27] and references therein. )…”

“…(ii) In the case K ≥ θ, some weaker estimates than (1.10) were stated in [26] and [27]. The proofs in [26] and [27] were based on uncertain pointwise estimates of (…”