Abstract

This paper is devoted to studying the long time behavior of solutions to a bipolar quantum hydrodynamic in one-dimensional space for general pressure functions. The model is usually applied to simulate some quantum effects in semiconductor devices. The decay rate for time variable is obtained by the entropy functional method and semidiscrete technique.

1. 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 [2–4]) and the Cahn-Hilliard equation. In the paper, we are mainly concerned with the following bipolar quantum drift-diffusion model in one-dimensional space:with the initial-boundary conditions as follows: where ,   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 is the Debye length.

Dolbeault et al. [1] studied the existence and uniqueness of the fourth-order parabolic equation 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.

As [6] has shown, by letting ,  ,  , and , we still borrow the semidiscrete system. Consider For the problem (1)–(4) and the semidiscrete system (5)–(8), we list some results (Theorems 1–3) which had been proved in [6].

Theorem 1. Assume , , and Let    be nondecreasing and assume that there exist two constants and such that Then there exists a weak solution to (1)–(4) such that

Theorem 2. Under the assumptions of Theorem 1, there exists a weak solution of (5)–(8) satisfying and in .

By defining the approximate solutions for , , [6] gave the following convergence results.

Theorem 3. Under the assumptions of Theorem 1, there exists a subsequence of (not relabeled) such that and it holdsfor all , where and is the duality product between and .

The main result of the paper is as follows.

Theorem 4. Under the assumptions of Theorem 1, let and let be the weak solution to (1)–(4). Then for , where the constants and only depend on , , and .

Here, we need the condition 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.

2. Semidiscrete Solutions

Introduce some discrete entropiesfor . For the positive entropies and , we have the following iteration estimate.

Lemma 5. Assume ,  . Then

Proof. Multiplying (5) and (6) by and , respectively, we have A direct calculation yields and similarly On the other hand, a simple calculation gives and similarly Letting be a test function in the Poisson equation (7), we get Combining (18)–(20) with (17), we haveFurthermore, the inequality for all impliesand similarly .  Now we have completed the proof.

Lemma 6. Assume , . Then

Proof. Multiply (5) and (6) by and , respectively, to get Using as a test function in (7) and applying the inequality   for  all  , we have The inequality   for   yieldsand similarly . By (24)–(27), we obtain Hence, (23) has been proved.

Lemma 7. Assume . Then

Proof. The inequality for gives and Jensen’s inequality yields The assertion finishes the proof of the lemma.

3. Exponential Decay

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

Lemma 8. Assume the function ,, in and ,, on . Then

Lemma 9. Assume the function , , in . Then

Lemma 10 (Criszar-Kullback-type inequality). Assume the function and . Then

Proof of Theorem 4. According to Lemmas 5, 8, 9, and 7, we get and then, by iterating the above inequality, we deduce that Moreover, for , we have Introduce the following functions: Equation (37) implies By applying Theorem 3, we conclude that there exists a subsequence of such that and a.e. in . Furthermore, we have and a.e. in . On the other hand, is bounded uniformly in from (39) and then Lebesgue’s convergence theorem yields for . Therefore, we have Applying Lemma 10, we have where and are both positive constants. Finally, multiply (7) by to get and then By letting , we can get the result of the theorem.