Long Time Decay Rate to a Bipolar Quantum Drift-Diffusion Model

By performing relaxation time limit in the quantumhydrodynamic equation, the semiconductor quantum drift-diffusion model can be obtained. Usually, it is applied to simulate the quantum effects, for example, resonant tunneling in semiconductor devices. Formally, the model also belongs to the field of the fourth-order parabolic equations (see [1]) including the thin film equation (see [2–4]) and the CahnHilliard equation. In the paper, we aremainly concernedwith the following bipolar quantum drift-diffusion model in onedimensional space:


Introduction
By performing relaxation time limit in the quantum hydrodynamic equation, the semiconductor quantum drift-diffusion model can be obtained.Usually, it is applied to simulate the quantum effects, for example, resonant tunneling in semiconductor devices.Formally, the model also belongs to the field of the fourth-order parabolic equations (see [1]) including the thin film equation (see [2][3][4]) and the Cahn-Hilliard equation.In the paper, we are mainly concerned with the following bipolar quantum drift-diffusion model in onedimensional space: 2   =  −  −  () , (, ) ∈   = Ω × (0, ) with the initial-boundary conditions as follows: where Ω = (0, 1),   is a constant,  is the electron density,  is the positively charged ion (or hole) density, and  is the electron static potential.  and   are the pressure functions and the function () is the doping profile.The parameter  is the scaled Plank constant and  > 0 is the Debye length.Dolbeault et al. [1] studied the existence and uniqueness of the fourth-order parabolic equation   + ((log)  )  = 0 with periodic boundary conditions.For the same equation, Jüngel and Toscani [5] used the entropy functional method and the semidiscrete technique to construct an iteration and obtained the exponential decay results.By employing the semidiscrete method, Jüngel and Violet obtained the existence of weak solution and gave the quasineutral limit in [6] to the bipolar quantum drift-diffusion model.
Generally, the bipolar model is more meaningful in physics and we will treat the case with a general pressure function.By applying the entropy method (see [7]) and iteration procedure which have already been used successfully in [5], we will get the long time exponential decay rate to the quantum drift-diffusion model ( 1)- (4).It is a key to deal with the coupling relationship in the Poisson equation (3).Moreover, since the maximum principle does not hold again for the high order partial differential equations, we need to overcome this difficulty for the purpose of getting uniform energy estimates.
The main result of the paper is as follows.
Here, we need the condition () ≡ 0 for the purpose of integration by parts and nonpositivity for some terms.
The paper is arranged as follows.We will prove some auxiliary lemmas at first in Section 2. The exponential decay rate will be established in Section 3.

Semidiscrete Solutions
Introduce some discrete entropies for  ∈ .For the positive entropies   and   , we have the following iteration estimate.
Proof.The inequality  − log ≥ /2 for  > 0 gives and Jensen's inequality yields The assertion finishes the proof of the lemma.

Exponential Decay
In order to prove Theorem 4, we list some known results (see [5]) at first.