Asymptotic stability of stationary solutions to the drift-diffusion model with the fractional dissipation
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Journal of Evolution Equations
Asymptotic stability of stationary solutions to the drift-diffusion model with the fractional dissipation Yuusuke Sugiyama
and Masakazu Yamamoto
Abstract. We study the drift-diffusion equation with fractional dissipation (−Δ)θ/2 arising from a model of semiconductors. First, we prove the existence and uniqueness of the small solution to the corresponding stationary problem in the whole space. Moreover, it is proved that the unique solution of non-stationary problem exists globally in time and decays exponentially, if initial data are suitably close to the stationary solution and the stationary solution is sufficiently small.
1. Introduction We consider the following Cauchy problem for the drift-diffusion equation arising from a model of semiconductors: ⎧ θ/2 ⎪ ⎨ ∂t u + (−Δ) u − ∇ · (u∇ψ) = 0, − Δψ = u − g, (1) ⎪ ⎩ u(0, x) = u 0 (x), where n ∈ N, ∇ = (∂1 , . . . , ∂n ), (−Δ)θ/2 ϕ = F −1 [|ξ |θ F[ϕ]], Δ = nj=1 ∂ 2j and u = u(t, x) and ψ = ψ(t, x) are real-valued unknown functions and stand for the density of electrons and the potential of electromagnetic field in a semiconductor, respectively. The given function g = g(x) is an impurity doping profile. From a microscopic perspective, we can regard the dynamics of the charged particle in a semiconductor as a stochastic process. When 0 < θ < 2, the fractional Laplacian (−Δ)θ/2 generates the jumping process in the stochastic process and describes the diffusion of particles. One can consider that the fractional dissipation is suitable to describe the dynamics of electrons in a semiconductor, since the electrons in a semiconductor jumps from a dopant into others. For probabilistic properties of the fractional Laplacian, we refer to Metzler and Klafter [18]. The purpose of this paper is to prove the unique existence of the small stationary solution with the small doping profile g(x) and its exponential stability. Namely, we prove that if initial data u 0 are close to the sufficiently small Mathematics Subject Classification: 35B33, 34K20, 93D20 Keywords: Drift-diffusion equation, Anomalous diffusion, Fractional dissipation, Cauchy problem. This work was supported by JSPS KAKENHI Grant Numbers 19K14573 and 19K03560.
Y. Sugiyama and M. Yamamoto
J. Evol. Equ.
stationary u st , then (1) has a global unique solution and some norm of u(t, x) − u st (x) decays exponentially in time. We remark that the Lebesgue space L n/θ (Rn ) and the n/ p−θ homogeneous Besov space B˙ p,q (Rn ) are invariant function spaces under the scalθ θ ing that λ u(λ t, λx). In fact, if (u(t, x), ψ(t, x)) is a solution of (1) with g ≡ 0, then (λθ u(λθ t, λx), λθ−2 ψ(λθ t, λx)) is also solutions of the first and second equations of (1). Our existence and stability results of stationary and non-stationary problems 1−θ (Rn ) with are discussed in the critical spaces L n/θ (Rn ) with 1 < θ < 2 and B˙ n,1 1−θ 0 (Rn ) L n (Rn ) and B n ˙ ˙ 1−θ (Rn ). Main 0 < θ ≤ 1. We note that B˙ n,1 n,1 (R ) Hn theorems for the stationary and the non-stationary problems are stated in
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