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

Authors: Yingjie Zhu; Bo Liang; Xiting Peng
Journal: Journal of Function Spaces
 (2015)

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## 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.


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## Body

## 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:(1)nt+ε22nlog⁡nxxxx-Pnnxx+nVxx=0,(2)pt+ε22plog⁡pxxxx-Pppxx-pVxx=0,(3)λ2Vxx=n-p-Cx,x,t∈QT=Ω×0,Twith the initial-boundary conditions as follows: (4)n=p=1,nx=px=0,V=VD,on∂Ω=0,1,n·,0=n0,p·,0=p0,inΩ,where Ω=(0,1),  VD is a constant, n is the electron density, p is the positively charged ion (or hole) density, and V is the electron static potential. Pn and Pp are the pressure functions and the function C(x) is the doping profile. The parameter ɛ is the scaled Plank constant and λ>0 is the Debye length.Dolbeault et al. [1] studied the existence and uniqueness of the fourth-order parabolic equation nt+(n(logn)xx)xx=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.As [6] has shown, by letting τ=T/n,  nk=n(x,kh),  pk=p(x,kh), and k=1,2,…,n, we still borrow the semidiscrete system. Consider (5)1τnk-nk-1+ε22nklog⁡nkxxxx-Pnnkxx+nkVkxx=0,(6)1τpk-pk-1+ε22pklog⁡pkxxxx-Pppkxx-pkVkxx=0,(7)λ2Vkxx=nk-pk-Cx,inΩ,(8)nk=pk=1,nkx=pkx=0,Vk=VD,on∂Ω.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.
AssumeC(x)∈L∞(Ω), 0≤n0,p0∈L1(Ω), and (9)∫Ωn0-log⁡n0dx+∫Ωp0-log⁡p0dx+∫Ωn0log⁡n0-1+1dx+∫Ωp0log⁡p0-1+1dx<∞.Let Pi∈C1[0,∞)  (i=n,p) be nondecreasing and assume that there exist two constants 0<q<7/2 and CP>0 such that Pix≤CP(1+xq)forx≥0,i=n,p. Then there exists a weak solution (n,p,V) to (1)–(4) such that (10)n,p≥0,inQT,n,p∈L7/2QT,log⁡n,log⁡p∈L20,T;H02Ω,nt,pt∈L10,T;H-3Ω,V∈L∞0,T;H2Ω.Theorem 2.
Under the assumptions of Theorem1, there exists a weak solution (nk,pk,Vk)∈H2(Ω)×H2(Ω)×H2(Ω) of (5)–(8) satisfying (lognk,logpk,Vk-VD)∈H02(Ω)×H02(Ω)×H01(Ω) and nk,pk>0 in Ω.By defining the approximate solutions(11)nτx,t=nkx,nτ~x,t=nk-1x,pτx,t=pkx,pτ~x,t=pk-1x,Vτx,t=Vkxfor x∈Ω, t∈((k-1)τ,kτ], [6] gave the following convergence results.Theorem 3.
Under the assumptions of Theorem1, there exists a subsequence of (n(τ),p(τ),V(τ)) (not relabeled) such that (12)log⁡nτ⇀log⁡n,log⁡pτ⇀log⁡p,weaklyinL20,T;H2Ω,nτ⟶n,pτ⟶p,stronglyinL2QT,Pnnτ⟶Pnn,Pppτ⟶Ppp,stronglyinL1QT,1τnτ-nτ~⇀nt,weaklyinLs0,T;H-3Ω,1τpτ-pτ~⇀pt,weaklyinLs0,T;H-3Ω,Vτ⇀V,weaklyinL20,T;H1Ωand it holds(13)∫0T∂tn,ϕdt=-∬QTnlog⁡nxxϕxxdxdt+∬QTPnnϕxxdxdt+∬QTnVxϕxdxdt,∫0T∂tp,ϕdt=-∬QTplog⁡pxxϕxxdxdt+∬QTPppϕxxdxdt-∬QTpVxϕxdxdt,λ2∬QTVxϕxdxdt=-∬QTn-pϕdxdtfor all ϕ∈C0∞(QT), where s=min{7/2q,14/11}>1 and ·,· is the duality product between H-3(Ω) and H03(Ω).The main result of the paper is as follows.Theorem 4.
Under the assumptions of Theorem1, let C(x)≡0 and let (n,p,V) be the weak solution to (1)–(4). Then (14)nt-1L1Ω+pt-1L1Ω+Vt-VDH1Ω≤C1η0+ξ0e-C2tfor t≥0, where the constants C1 and C2>0 only depend on Ω, n0, and p0.Here, we need the conditionC(x)≡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 Section2. The exponential decay rate will be established in Section 3.


## 2. Semidiscrete Solutions

Introduce some discrete entropies(15)ηk=∫Ωnklog⁡nk-1+1dx,ξk=∫Ωpklog⁡pk-1+1dx,qk=∫Ωnk-log⁡nkdx,ζk=∫Ωpk-log⁡pkdx,for k∈N. For the positive entropies ηk and ξk, we have the following iteration estimate.Lemma 5.
Assumeηk-1,ξk-1<∞,  k≥1. Then (16)1τηk-ηk-1+1τξk-ξk-1+ε22∫Ωnklog⁡nkxx2dx+ε22∫Ωpklog⁡pkxx2dx≤0.Proof.
Multiplying (5) and (6) by lognk and logpk, respectively, we have (17)1τ∫Ωnk-nk-1log⁡nkdx+ε22∫Ωnklog⁡nkxx2dx=-∫ΩPnnkx-nkVkxlog⁡nkxdx,1τ∫Ωpk-pk-1log⁡pkdx+ε22∫Ωpklog⁡pkxx2dx=-∫ΩPppkx+pkVkxlog⁡pkxdx.
A direct calculation yields(18)-∫ΩPnnkxlog⁡nkxdx=-∫ΩPn′nknkx2nkdx≤0and similarly -∫Ω(Pp(pk))x(logpk)xdx≤0. On the other hand, a simple calculation gives (19)∫ΩnkVkxlog⁡nkxdx=∫ΩVkxnkxdx,and similarly -∫ΩpkVkx(logpk)xdx=-∫ΩVkxpkxdx. Letting nk-pk be a test function in the Poisson equation (7), we get (20)λ2∫ΩVkxnk-pkxdx=-∫Ωnk-pk2dx≤0.Combining (18)–(20) with (17), we have(21)1τ∫Ωnk-nk-1log⁡nkdx+ε22∫Ωnklog⁡nkxx2dx+1τ∫Ωpk-pk-1log⁡pkdx+ε22∫Ωpklog⁡pkxx2dx≤0.
Furthermore, the inequality-logx+x-1≥0 for all x>0 implies(22)1τ∫Ωnk-nk-1log⁡nkdx=1τηk-ηk-1+1τ∫Ωnk-1-log⁡nknk-1+nknk-1-1dx≥1τηk-ηk-1and similarly 1/τ∫Ω(pk-pk-1)logpkdx≥(1/τ)(ξk-ξk-1).  Now we have completed the proof.Lemma 6.
Assumeqk-1,ζk-1<∞, k≥1. Then (23)1τqk-qk-1+ε22∫Ωlog⁡nkxx2dx+1τζk-ζk-1+ε22∫Ωlog⁡pkxx2dx≤0.Proof.
Multiply (5) and (6) by 1-1/nk and 1-1/pk, respectively, to get (24)1τ∫Ωnk-nk-1-1+nk-1nkdx+ε22∫Ωlog⁡nkxx2dx=ε22∫Ωlog⁡nkxxlog⁡nkx2dx-∫ΩPn′nknkx2nk2dx+∫ΩVkxlog⁡nkxdx=ε22∫Ωlog⁡nkx3x2dx-∫ΩPn′nknkx2nk2dx+∫ΩVkxlog⁡nkxdx≤∫ΩVkxlog⁡nkxdx,(25)1τ∫Ωpk-pk-1-1+pk-1pkdx+ε22∫Ωlog⁡pkxx2dx≤-∫ΩVkxlog⁡pkxdx.
Usinglognk-logpk as a test function in (7) and applying the inequality (x1-x2)(logx1-logx2)≥0  for  all  x1,x2>0, we have (26)λ2∫ΩVkxlog⁡nk-log⁡pkxdx=-∫Ωnk-pklog⁡nk-log⁡pkdx≤0.The inequality x-1≥logx  for  x>0 yields(27)∫Ωnk-nk-1-1+nk-1nkdx=qk-qk-1+∫Ω-lognk-1nk+nk-1nk-1dx≥qk-qk-1and similarly ∫Ω(pk-pk-1-1+pk-1/pk)dx≥ζk-ζk-1. By (24)–(27), we obtain (28)1τqk-qk-1+ε22∫Ωlog⁡nkxx2dx+1τζk-ζk-1+ε22∫Ωlog⁡pkxx2dx≤∫ΩVkxlog⁡nk-log⁡pkxdx≤0.Hence, (23) has been proved.Lemma 7.
Assumeq0+ζ0<∞. Then (29)0<e-q0+ζ0≤∫Ωnk+pkdx≤2q0+ζ0<∞.Proof.
The inequalityx-logx≥x/2 for x>0 gives (30)∫Ωnk+pkdx≤2qk+ζk≤2q0+ζ0and Jensen’s inequality yields (31)-log⁡∫Ωnk+pkdx≤-∫Ωlog⁡nk+pkdx+∫Ωnk+pkdx≤12∫Ωnk+pkdx≤q0+ζ0<∞.The assertion finishes the proof of the lemma.


## 3. Exponential Decay

In order to prove Theorem4, we list some known results (see [5]) at first.Lemma 8.
Assume the functionu∈H2(Ω),u>0, in Ω and u=1,ux=0, on ∂Ω. Then (32)∫Ωulog⁡uxx2dx≥8u-1L∞Ω2.Lemma 9.
Assume the functionu∈L∞(Ω), u>0, in Ω. Then (33)∫Ωulog⁡u-1+1dx≤∫Ωudx+2u-1L∞Ω2.Lemma 10 (Criszar-Kullback-type inequality).
Assume the function0<u∈L1(Ω) and q=∫Ω(u-logu)dx<∞. Then (34)∫Ωulog⁡u-1+1dx≥11+2q2∫Ωu-1dx2.Proof of Theorem4.
According to Lemmas5, 8, 9, and 7, we get (35)
η
k
-
η
k
-
1
+
ξ
k
-
ξ
k
-
1
≤
-
τ
2
ε
2
∫
Ω
n
k
log
⁡
n
k
x
x
2
d
x
+
ε
2
2
∫
Ω
p
k
log
⁡
p
k
x
x
2
d
x
≤
-
4
τ
ε
2
n
k
-
1
L
∞
Ω
2
+
p
k
-
1
L
∞
Ω
2
≤
-
4
τ
ε
2
η
k
∫
Ω
n
k
d
x
+
2
+
ξ
k
∫
Ω
p
k
d
x
+
2
≤
-
2
τ
ε
2
q
0
+
ζ
0
+
1
η
k
+
ξ
kand then, by iterating the above inequality, we deduce that (36)ηk+ξk≤1+2τε2q0+ζ0+1-1ηk-1+ξk-1≤1+2τε2q0+ζ0+1-kη0+ξ0.Moreover, for (k-1)τ<t≤kτ, we have (37)ηk+ξk≤1+2τε2q0+ζ0+1-t/τη0+ξ0.Introduce the following functions: (38)ητt=∫Ωnτtlog⁡nτt-1+1dx,ξτt=∫Ωpτtlog⁡pτt-1+1dx.Equation (37) implies (39)ητt+ξτt≤η0+ξ01+2τε2q0+ζ0+1-t/τ.
By applying Theorem3, we conclude that there exists a subsequence of (n(τ),p(τ)) such that n(τ)(t)→n and p(τ)(t)→p a.e. in Ω. Furthermore, we have n(τ)logn(τ)→nlogn and p(τ)logp(τ)→plogp a.e. in Ω. On the other hand, η(τ)+ξ(τ) is bounded uniformly in τ from (39) and then Lebesgue’s convergence theorem yields η(τ)+ξ(τ)→η+ξ for t∈(0,T). Therefore, we have (40)ηt+ξt≤η0+ξ0limτ→0⁡1+2τε2q0+ζ0+1-t/τ=η0+ξ0e-2/q0+ζ0+1t.Applying Lemma 10, we have (41)nt-1L1Ω+pt-1L1Ω≤C1η0+ξ0e-C2t,where C1 and C2 are both positive constants. Finally, multiply (7) by Vk-VD to get (42)λ2∫ΩVk-VDx2dx=-∫Ωnk-pkVk-VDdx≤∫Ωnk-pkdxVk-VDL∞Ω≤CVk-VDH1Ω∫Ωnk-pkdxand then (43)λ2Vτ-VDH1Ω≤C∫Ωnτ-pτdx.By letting τ→0, we can get the result of the theorem.



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*Source: 163630-2015-06-18.xml*