DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2014.19.1601DYNAMICAL SYSTEMS SERIES BVolume 19, Number 6, August 2014 pp. 1601–1626

ANALYSIS ON THE INITIAL-BOUNDARY VALUE PROBLEM

OF A FULL BIPOLAR HYDRODYNAMIC MODEL FOR

SEMICONDUCTORS

Haifeng Hu and Kaijun Zhang ∗

School of Mathematics and Statistics, Northeast Normal University

Changchun, MO 130024, China

(Communicated by Yuan Lou)

Abstract. We consider the initial boundary value problem of the one dimen-

sional full bipolar hydrodynamic model for semiconductors. The existence and

uniqueness of the stationary solution are established by the theory of stronglyelliptic systems and the Banach fixed point theorem. The exponentially as-

ymptotic stability of the stationary solution is given by means of the energy

estimate method.

1. Introduction. In this paper, we consider the following 1-D full bipolar hydro-dynamic (HD) model for semiconductors:

nit + jix = 0, (1a)

jit +( j2

i

ni+ niθi

)x

= (−1)i−1niφx − ji, (1b)

niθit + jiθix +2

3

( jini

)xniθi −

2

3θixx =

1

3

j2i

ni− ni(θi − θ), (1c)

φxx = n1 − n2 −D(x), i = 1, 2, ∀(t, x) ∈ R+ × Ω, (1d)

where R+ is the set of nonnegative real number, Ω := (0, 1) ⊂ R. The unknownvariables ni, ji, θi are the electron (i = 1) and the hole (i = 2) density, the current ofthe electron and the current of the hole, the absolute temperature of the electron andthe absolute temperature of the hole, respectively. The unknown variable φ is theelectrostatic potential. The positive constants θ means ambient device temperature.The doping profile D is a given function, which represents the density of impuritiesin semiconductors. The assumption that the pressure obey the Boyle-Charles lawpi(ni, θi) = niθi has been used in the system. Comparing with the isentropic orunipolar case, this model is a more physical case.

Based on the mini-size characterization of semiconductor devices, it is moremeaningful to study the model over the bounded domain with physically admis-sible boundary conditions. For this reason, we consider the IBVP of (1). Therelated initial and boundary data are given by

(ni, ji, θi)(0, x) = (ni0, ji0, θi0)(x), (2)

2010 Mathematics Subject Classification. 35B40, 35M13.Key words and phrases. Full hydrodynamic model, energy estimate, asymptotic stability, sta-

tionary solution, strongly elliptic systems.∗: the corresponding author; Zhang’s work is partially supported by the NSFC (No.11371082).

1601

1602 HAIFENG HU AND KAIJUN ZHANG

and

ni(t, 0) = nil > 0, ni(t, 1) = nir > 0, (3)

θi(t, 0) = θil > 0, θi(t, 1) = θir > 0, (4)

φ(t, 0) = 0, φ(t, 1) = φr > 0. (5)

where nil, nir, θil, θir and φr are positive constants. In physics, (3)-(5) are calledOhmic contact boundary conditions. Furthermore, the initial data is supposed tobe compatible with boundary data

ni0(0) = nil, ni0(1) = nir, θi0(0) = θil, θi0(1) = θir, ji0x(0) = ji0x(1) = 0. (6)

The mathematical structure of the model (1) is a quasi-linear hyperbolic-parabolic-elliptic coupled system, in which equation (1a) is the conservation law of mass,equation (1b) is the balance law of momentum coupled to the Poisson equation(1d) for the electric potential, and heat equation (1c) follows from the balance lawof energy. We will be interested in the solvability of this model in the region wherethe properties

infx∈Ω

S[ni, ji, θi] > 0, S[ni, ji, θi] := θi −j2i

n2i

, (7a)

infx∈Ω

ni > 0, (7b)

infx∈Ω

θi > 0, (7c)

are satisfied. Conditions (7a) are called subsonic conditions. Conditions (7b) and(7c) are positivity of the density and the temperature, respectively. From the sub-sonic conditions, we can derive the characteristic speeds of the hyperbolic equations(1a)-(1b) as follows

λi1 = ui −√θi < 0, λi2 = ui +

√θi > 0, (8)

where ui := ji/ni means the velocity of electron flow (i = 1) and hole flow (i =2). This leads to the linearized system of (1a)-(1b) is a symmetric hyperbolicsystem. Therefore, we can see that the boundary conditions (3)-(5) are sufficientand necessary for the well-posedness of the initial boundary value problem.

The hydrodynamic (HD) models are introduced firstly by Bløtekjær [2], whichare used to describe the charged fluid particles such as electrons and holes in semi-conductor devices. For mathematical derivations, the books [14, 21] are also goodreferences.

In the unipolar case, Luo, Natalini and Xin [17] first studied the large timebehavior of the solutions to the Cauchy problem of the 1-D HD model with flatdoping profile in the switch-off case (the current at far fields is zero), and theychose the asymptotic profile as the stationary solution of the model. Huang, Mei,Wang and Yu [12, 13] improved the previous result, they considered the switch-oncase, and obtained the exponentially asymptotic stability of the stationary wavein 1-D case and the planar stationary wave in 3-D case. Ali, Bini and Rionero [1]proved the stability of the stationary solution of full HD model with non-flat dopingprofile over R, and they assume that the stationary solution satisfies j ≡ 0 andθ ≡ θ. Regarding the bounded domain case, Degond and Markowich [3] studied theexistence and uniqueness of the stationary solution of the 1-D HD model, in whichthe current density j is given. Nishibata and Suzuki [24] reconsidered the existenceand uniqueness for a given boundary voltage φr rather than a given current density.

A FULL BIPOLAR HD MODEL FOR SEMICONDUCTORS 1603

The asymptotic stability of the stationary solution of the HD model is proved by Li,Markowich and Mei [18] with flat doping profile |D(x) − nl| 1. The asymptoticstability with non-flat doping profile is proved by Guo and Strauss [5] (also see[24]). For the full HD model, Hsiao and Wang [8] proved the asymptotic stabilityof the stationary solution with flat doping profile in 1-D case, in which j ≡ 0 andθ ≡ θ. Nishibata and Suzuki [25] showed the existence, uniqueness and asymptoticstability of the stationary solution with non-flat doping profile, and also in 1-D case,the stationary solution does not have to be j ≡ 0 and θ ≡ θ, and the density nchanges steeply. Hsiao, Jiang and Zhang [7] established the asymptotic stability

of the constant steady state n ≡ 1, j ≡ 0 and θ ≡ θ = 1 with flat doping profileD(x) ≡ 1 in 3-D case.

In the bipolar case, the related study seems very limited so far due to the complex-ity of structure of the model. Tsuge [28] considered the existence and uniquenessof the stationary solution of 1-D HD model with non-flat doping profile and thespecial Ohmic contact boundary conditions, the restriction of the boundary datais used to overcome the bipolar coupling effect. Natalini [23] and Zhang [31] es-tablished the global existence and zero relaxation limit results of weak solutions of1-D HD model by the theory of compensated compactness on the whole space Rand bounded domain (0, 1), respectively. Li [19] proved the unique existence andasymptotic stability of the stationary solutions of 1-D HD model with completelyflat doping profile D(x) ≡ 0, the same pressure functions and the special boundarydata.These restrictive assumptions are also used to overcome the bipolar couplingeffect. Mei, Rubino and Sampalmieri [22] showed the asymptotic stability of thestationary solution of the 1-D HD model with two different pressure functions andgeneral boundary conditions. Recently, Donatelli, Mei, Rubino and Sampalmieri [4]studied the Cauchy problem for 1-D HD model with two different pressure functionsand a non-flat doping profile. They showed the algebraically asymptotic stabilityof the stationary solution. The optimal algebraic convergence rates are obtained.

In the present paper, we are mainly concerned with the existence, uniqueness andasymptotic stability of the stationary solution of (1)-(5) with the general bound-ary conditions and the relatively general doping profile. To our best knowledge,the obtained result here is the first one for the mathematical analysis of the initialboundary value problem of the full bipolar HD model in 1-D case. In order to over-come the difficulties arising from the bipolar coupling effect under consideration,we first apply the theory of strongly elliptic systems and the Banach fixed pointargument to solve the stationary problem rather than employing the traditionalLeray-Schauder fixed point argument because it does not work to get the boundof the stationary densities in terms of the maximum principle. As far as the ex-ponentially asymptotic stability of the stationary solution is concerned, we observethat there is a loss of the partial dissipation rate in the basic energy estimate (92).Namely, we can’t directly derive the estimate of the perturbed densities ψ1, ψ2 bythe electric field σx (see Remark 1). To overcome this difficulty, we have to exploredeeply the special structure of the perturbed equations (110) and (113). Firstly, wefind that the spatial derivatives of the perturbed variables can still be controlledby the time derivatives with the help of the perturbed equations (110) and (113)in our bipolar case (see (122)). In addition, we can establish the higher order es-timate (123) via the perturbed equations (110), and then can compensate the lossof the dissipation rate in the basic estimate by virtue of the partial dissipation rate∫ 1

0

∑2i=1 θin

−1i ψ2

ixdx in (123)|k=0 and the Poincare inequality (see (153)). We claim

1604 HAIFENG HU AND KAIJUN ZHANG

that this remedy is available for the purpose that we want to close the uniformestimate (85). In fact, we can define the total energy E(t) and its dissipation rate

F (t) by (151) and (152) respectively, and find that F (t) can be controlled from be-

low by E(t) based on the remedy (153) and the estimate (122). Although we can’t

show that E(t) is equivalent to F (t) as the unipolar case, it is enough to achieve ourpurpose in bipolar case. By the way, we can reconsider the problems in [22, 19] withthe general boundary conditions (3) and (5), with two different pressure functionsand with the relatively general doping profile D, and then obtain the same resultsin [22, 19] by our methods developed in this paper.

Before stating the main results, we introduce some notations. For l, k ∈ N0,H l(Ω) denotes the l-th order Sobolev space in the L2 sense, equipped with thenorm ‖ · ‖l. We note H0 = L2 and ‖ · ‖ := ‖ · ‖0. Ck([0, T ];H l(Ω)) denotes thespace of the k-times continuously differentiable functions on the interval [0, T ] withvalues in H l(Ω). Hk(0, T ;H l(Ω)) is the space of Hk-functions on (0, T ) with valuesin H l(Ω). Bk(Ω) denotes the space of the functions whose derivatives up to k-thorder are continuous and bounded over Ω, equipped with the norm

|f |k :=

k∑i=0

supx∈Ω

|∂ixf(x)|.

In order to show the asymptotic stability of the stationary solution, it is conve-nient to introduce the function spaces

Xkl ([0, T ]) := ∩li=0Ci([0, T ];Hk+l−i(Ω)),

Xl([0, T ]) := X0l ([0, T ]) for l, k = 0, 1, 2,

Y([0, T ]) := C([0, T ];H2(Ω)) ∩ C1([0, T ];L2(Ω)).

The strength parameter δ describing the magnitude of the stationary solutionand the doping profile is given as follows

δ := ‖D − d‖1 +

2∑i=1

(|nir − nil|+ |θil − θ|+ |θir − θ|

)+ |φr|, (9)

where

d := n1l − n2l, (10)

Let C denote the generic positive constant, and C(α, β, · · · ) or Cα,β,··· the positiveconstant depending on α, β, · · · .

Theorem 1.1. Let D ∈ H1(Ω), for arbitrary n1l > 0, n2l > 0 and θ > 0,there exists δ1 > 0 such that if δ ≤ δ1, then there is a unique stationary solu-tion (n1, j1, θ1, n2, j2, θ2, φ) ∈ [B2(Ω)]6 × B4(Ω) of the IBVP (1)-(5), satisfying theconditions (7) and the estimates as follows

1

2nil ≤ ni(x) ≤ 2nil,

1

2θ ≤ θi(x) ≤ 2θ, ∀x ∈ Ω, i = 1, 2, (11a)

|φ|4 +

2∑i=1

(|ji|+ |ni − nil|2 + |θi − θ|2

)≤ C(n1l, n2l, θ)δ. (11b)

The asymptotic stability of the stationary solution is summarized in the followingtheorem.

A FULL BIPOLAR HD MODEL FOR SEMICONDUCTORS 1605

Theorem 1.2. Suppose that the initial data ni0, ji0, θi0 ∈ H2(Ω) and satisfy (7),

and the boundary data nil, nir, θil, θir, φr satisfy (3)-(6), i = 1, 2. Let (n1, j1, θ1,

n2, j2, θ2, φ) be the stationary solution in Theorem 1.1. Then there exits ε > 0

such that if δ +∑2i=1 ‖(ni0 − ni, ji0 − ji, θi0 − θi)‖2 ≤ ε, then the IBVP (1)-

(5) has a unique global subsonic solution (n1, j1, θ1, n2, j2, θ2, φ) satisfying ni ∈X2([0,∞)), ji ∈ X1

1([0,∞)), θi ∈ Y([0,∞)) ∩ H1loc(0,∞;H1(Ω)), i = 1, 2 and φ ∈

C2([0,∞);H2(Ω)). Moreover, the global solution verifies the additional regularity

jitt ∈ L2loc(0,∞;L2(Ω)) and φ− φ ∈ X2

2([0,∞)), and the decay estimate

2∑i=1

‖(ni − ni, ji − ji, θi − θi)(t)‖2 + ‖(φ− φ)(t)‖4

≤ C2∑i=1

‖(ni0 − ni, ji0 − ji, θi0 − θi)‖2e−γt, (12)

where C and γ are positive constants independent of time variable t.

The idea of the proof is outlined as follows. Firstly, we show the existence anduniqueness of the stationary solution by the theory of strongly elliptic systems andthe Banach fixed-point theorem. In this procedure, we observed that the strengthparameter δ = 0 (switch-off case) implies the existence and uniqueness of a constantstationary solution (n1l, 0, θ, n2l, 0, θ, 0). If 0 < δ 1 (switch-on case), then thecorresponding stationary solution can be regarded as a regular perturbation aroundthe constant stationary solution. Secondly, we prove the local existence of (1)-(5)by the iteration method similar to [25]. The solvability of linearized problem of(1)-(5) is shown by the Galerkin method (cf. [27, 30]) and the theory of symmetrichyperbolic system (cf. Theorem-A1 in [26]).Then we can define a solution oper-ator S2 by the solvability of this linearized problem and introduce a function setX(T ;m,M). We can suitably choose the constants T0,m0,M0 which depend onthe initial data (2) by the similar way in [15, 16, 25] such that X(T0;m0,M0) isinvariant under the operator S2. Eventually, we can prove S2 has a unique fixedpoint in the set X(T0;m0,M0) by the iteration method, which is the desired localsolution. Finally, because Theorem 1.2 can be proved by the continuation argumentbased on the local existence and the uniform a priori estimate, the only thing leftis to establish the uniform a priori estimate.

This paper is organized as follows. In Sect.2, we make a detailed discussion withthe proof of the existence and uniqueness of the stationary solution. In Sect.3.1, wegive the local existence theorem of (1)-(5). In Sect.3.2, we obtain the asymptoticstability of the stationary solution.

2. Existence and uniqueness of a stationary solution. This section is devotedto the proof of the existence and uniqueness of a stationary solution. We constructthe stationary solution by applying the standard L2-theory of the boundary valueproblem for strongly elliptic system (see Chapter 4 in [20], also see [29]) and theBanach fixed-point theorem (see Theorem 5.1 in [6]). The main point is that weregard the stationary problem with 0 < δ 1 as a regular perturbation problem ofthe stationary problem with δ = 0.

The stationary problem corresponding to the original problem (1)-(5) is the fol-lowing system of equations

1606 HAIFENG HU AND KAIJUN ZHANG

jix = 0, (13a)( j2i

ni+ niθi

)x

= (−1)i−1niφx − ji, (13b)

jiθix +2

3

( jini

)xniθi −

2

3θixx =

1

3

j2i

ni− ni(θi − θ), (13c)

φxx = n1 − n2 −D(x), (13d)

with the same boundary data as in (3)-(5)

ni(0) = nil > 0, ni(1) = nir > 0, (14)

θi(0) = θil > 0, θi(1) = θir > 0, (15)

φ(0) = 0, φ(1) = φr > 0. (16)

If δ = 0, then the stationary problem (13)-(16) reduce to

jix = 0, (17a)( j2i

ni+ niθi

)x

= (−1)i−1niφx − ji, (17b)

jiθix +2

3

( jini

)xniθi −

2

3θixx =

1

3

j2i

ni− ni(θi − θ), (17c)

φxx = n1 − n2 − d, (17d)

with the boundary data

ni(0) = ni(1) = nil > 0, θi(0) = θi(1) = θ > 0, (18)

φ(0) = φ(1) = 0. (19)

One can easily see that there is a unique constant solution (n1l, 0, θ, n2l, 0, θ, 0) of(17)-(19), which is exactly the subsonic stationary flow. For 0 ≤ δ 1, we introducethe regular perturbation variables as follows

nδi (x) := ni(x)− nil, θδi (x) := θi(x)− θ,

jδi := ji − 0 = ji, φδ(x) := φ(x)− 0 = φ(x). (20)

In particular, we have (n01, j

01 , θ

01, n

02, j

02 , θ

02, φ

0) = (0, 0, 0, 0, 0, 0, 0).An explicit formula of the electrostatic potential

φ(x) = Φ[n1, n2](x)

:=

∫ x

0

∫ y

0

(n1 − n2 −D)(z)dzdy +

(φr −

∫ 1

0

∫ y

0

(n1 − n2 −D)(z)dzdy

)x. (21)

follows from (13d) and (16).From (13b) and (16), we have the following current-voltage relationships

(−1)i−1φr = F [nir, ji, θir]− F [nil, ji, θil]−∫ 1

0

θix ln nidx+ ji

∫ 1

0

n−1i dx, (22)

where F [a, b, c] := b2

2a2 + c+ c ln a.

A FULL BIPOLAR HD MODEL FOR SEMICONDUCTORS 1607

From (22), we can write ji explicitly by the formula

ji = Ji[ni, θi] := 2(Bib +

∫ 1

0

θix ln nidx)Ki[ni, θi]

−1, (23)

Ki[ni, θi] :=

∫ 1

0

n−1i dx

+

√(∫ 1

0

n−1i dx

)2

+ 2(Bib +

∫ 1

0

θix ln nidx)(n−2ir − n

−2il

),

Bib := (−1)i−1φr − θir + θil − θir lnnir + θil lnnil,

under the subsonic conditions. It is easy to see that the formula (23) is well definedwhen δ 1.

By virtue of the formulas (21) and (23), we can firstly find out the solutions

ni, θi, and then define a constant ji := Ji[ni, θi] and a function φ := Φ[n1, n2].

Obviously, (n1, j1, θ1, n2, j2, θ2, φ) is the desired subsonic stationary solution of theBVP (13)-(16). Therefore, dividing (13b) by ni and differentiating the resultantequality in x, we have

( 1

niS[ni, ji, θi]nix

)x

+ θixx + (−1)i(n1 − n2 −D(x)) = 0, (24a)

jiθix +2

3

( jini

)xniθi −

2

3θixx =

1

3

j2i

ni− ni(θi − θ), (24b)

with the boundary data

ni(0) = nil > 0, ni(1) = nir > 0, θi(0) = θil > 0, θi(1) = θir > 0. (25)

In the system (24), the constant ji = Ji[ni, θi] is a good non-local term. As men-tioned above, in the subsonic region, the unique solvability of (13)-(16) is equivalentto the unique solvability of the BVP (24)-(25), (21) and (23).

In the rest of this section, we focus on the unique solvability of (24)-(25). Thatis, we mainly present the following results.

Lemma 2.1. Under the assumptions of Theorem 1.1, the BVP (24)-(25) has a

unique solution ni, θi ∈ B2(Ω) satisfying the condition (7b),(7c) and

1

2nil ≤ ni(x) ≤ 2nil,

1

2θ ≤ θi(x) ≤ 2θ, ∀x ∈ Ω, i = 1, 2, (26a)

2∑i=1

(|ni − nil|2 + |θi − θ|2

)≤ C(n1l, n2l, θ)δ. (26b)

Proof. Step 1. Substituting (20) into BVP (24)-(25), we have

(1

nil + nδiS[nil + nδi , ji, θ + θδi ]n

δix

)x

+θδixx + (−1)i(nδ1 − nδ2 − (D(x)− d)

)= 0,

(27a)

jiθδix +

2

3

(ji

nil + nδi

)x

(nil + nδi

)(θ + θδi

)−2

3θδixx =

1

3

j2i

nil + nδi−(nil + nδi

)θδi ,

(27b)

1608 HAIFENG HU AND KAIJUN ZHANG

with the boundary data

nδi (0) = 0, nδi (1) = nir − nil, θδi (0) = θil − θ, θδi (1) = θir − θ. (28)

In order to extract the linear principal part of the system (27), we set

W δ :=(nδ1, j1, θ

δ1, n

δ2, j2, θ

δ2

)T, W 0 := (n1l, 0, θ, n2l, 0, θ)

T , W = W 0 +W δ,

and we formally write the system (27) as F(W ) = 0. By ddεF(W 0 + εW δ)|ε=0, we

obtain a 4× 4 system −(AUx)x +BU = f(U) + g(x), x ∈ Ω, (29a)

U |∂Ω = h(x), (29b)

where the unknown variable is

U(x) :=(nδ1, θ

δ1, n

δ2, θ

δ2

)T(x), (30)

the coefficient matrices is defined as

A :=

θ/n1l 0 0 0

0 2/3 0 00 0 θ/n2l 00 0 0 2/3

, B :=

1 −3n1l/2 −1 00 n1l 0 0−1 0 1 −3n2l/20 0 0 n2l

, (31)

the nonlinearity is

f(U) := (f1, f2, f3, f4)T (U), (32)

where

f2i−1(U) := − 1

(nil + nδi )2S[nil + nδi , ji, θ + θδi ](n

δix)2

+1

nil + nδi

(θδixn

δix +

2j2i

(nil + nδi )3

(nδix)2)

+1

nil

(θδi n

δixx

− j2i

(nil + nδi )2nδixx

)+( 1

nil + nδi− 1

nil

)S[nil + nδi , ji, θ + θδi ]n

δixx

+3

2jiθ

δix −

θ + θδi2(nil + nδi )

jinδix −

j2i

2(nil + nδi )+

3

2nδi θ

δi , (33a)

f2i(U) := −jiθδix +θ + θδi

3(nil + nδi )jin

δix +

j2i

3(nil + nδi )− nδi θδi , i = 1, 2, (33b)

and ji = Ji[nil + nδi , θ + θδi ]. The nonhomogeneous term is defined as

g(x) := (D(x)− d, 0,−(D(x)− d), 0)T , (34)

and the boundary data is given by

h(x) :=(h1, h2, h3, h4)T (x), (35a)

h2i−1(x) :=(nir − nil)x, (35b)

h2i(x) :=(θir − θil)x+ θil − θ. (35c)

We set

λ := min θ

n1l,θ

n2l,

2

3

> 0, (36)

then one can see that

Aξ · ξ ≥ λ|ξ|2, ∀ξ ∈ R4. (37)

A FULL BIPOLAR HD MODEL FOR SEMICONDUCTORS 1609

Namely, the linear operator L defined by

LU := −(AUx)x +BU, (38)

is a second-order strongly elliptic operator.Step 2. In order to apply the Fredholm alternative result in the theory of stronglyelliptic system to define a solution operator S1, let us consider the homogeneousproblem as follows

LU(x) = 0, x ∈ Ω, (39a)

U(x) = 0, x ∈ ∂Ω. (39b)

In terms of the energy method, (39) only owns the zero solution,namely

U ≡ 0. (40)

According to the Fredholm alternative result, we can conclude that the nonhomo-geneous problem

LU(x) = f(x), x ∈ Ω, (41a)

U(x) = h(x), x ∈ ∂Ω. (41b)

has a unique solution U . Furthermore, if ∂Ω is smooth, f ∈ H1(Ω) and h ∈ H3(Ω),then the solution U ∈ H3(Ω) satisfying the estimate

‖U‖3 ≤ C(‖f‖1 + ‖h‖3), (42)

where the positive constant C depends on the coefficients of L, namely, C =C(n1l, n2l, θ). The linear operator L−1

h denotes the well-defined inverse of the op-erator L with the Dirichlet boundary condition (41b). Now we can formally definea solution operator S1 : U 7→ V based on (29). Precisely, for a given vector-valuedfunction U , we can find a unique vector-valued function V =: S1U solves the bound-ary value problem −(AVx(x))x +BV (x) = f(U(x)) + g(x), x ∈ Ω, (43a)

V (x) = h(x), x ∈ ∂Ω, (43b)

where f(U), g(x) and h(x) are given by (32)-(35). That is

S1U := L−1h (f(U) + g). (44)

Formally, we can see that the fixed-point U of the operator S1 is the desired solutionof (29).Step 3. We want to prove that the operator S1 : W[N ] → W[N ] is a contrac-tion mapping, where the space W[N ] is a complete metric space properly selected.Through the analysis for the nonlinearity (33), the space is defined as

W[N ] :=W ∈ H3(Ω)

∣∣∣‖W‖3 ≤ Nδ, W |∂Ω = h

(45)

Here the positive constant N will be determined later. In fact, it follows from thetrace theorem that W[N ] is a closed subset of space H3(Ω) for any N > 0 and δ ≥ 0(Note that if δ = 0, then h ≡ 0 and W[N ] = 0). In addition, for any U ∈W[N ],we have f(U) ∈ H1(Ω), then S1U ∈ H3(Ω), S1U |∂Ω = h and the estimate

‖S1U‖3 ≤ C1(n1l, n2l, θ)(‖f(U) + g‖1 + ‖h‖3) (46)

follows from (45) and (42).In order to show that the operator S1 is onto, we have to estimate (46) as follows

‖S1U‖3 ≤ C1(n1l, n2l, θ)(‖f(U)‖1 + ‖g‖1 + ‖h‖3). (47)

1610 HAIFENG HU AND KAIJUN ZHANG

From (34), (35) and (9), we have

‖g‖1 + ‖h‖3 ≤ C2δ. (48)

By the similar way in [25], the estimate

|ji| ≤ C(nil, N)δ, (49)

follows from (23). Through the straightforward but tedious calculation, we canestablish the estimate

‖f(U)‖1 ≤ C3(n1l, n2l, θ, N)δ2, ∀U ∈W[N ], (50)

where we have used the Sobolev embedding theorem, (49) and (33). Substitutingthe estimates (48) and (50) into (47), we have

‖S1U‖3 ≤ C4(n1l, n2l, θ, N)δ2 + C5(n1l, n2l, θ)δ. (51)

Let

C4(n1l, n2l, θ, N)δ2 + C5(n1l, n2l, θ)δ ≤ Nδ. (52)

Since δ > 0, we have

C4(n1l, n2l, θ, N)δ + C5(n1l, n2l, θ) ≤ N. (53)

Note that 0 < δ 1, we can define N as

N := 2C5(n1l, n2l, θ) > 0. (54)

If we take 0 < δ ≤ C5/C4(n1l, n2l, θ, 2C5), then

S1U ∈W[N ], ∀U ∈W[N ]. (55)

Next, we prove that S1 is contractive. To this end, for any U1, U2 ∈ W[N ], weconsider

S1U1 − S1U2 = L−1h (f(U1) + g)− L−1

h (f(U2) + g)

= L−1h−h(f(U1) + g − (f(U2) + g))

= L−10 (f(U1)− f(U2)). (56)

From (42) and (56), we have

‖S1U1 − S1U2‖3 ≤ C1(n1l, n2l, θ)(‖f(U1)− f(U2)‖1 + ‖0‖3)

= C1(n1l, n2l, θ)‖f(U1)− f(U2)‖1. (57)

In order to complete the proof, we need to estimate ‖f(U1) − f(U2)‖1 as follows,for any

U1 =(nδ11, θ

δ11, n

δ21, θ

δ21

)T, U2 =

(nδ12, θ

δ12, n

δ22, θ

δ22

)T∈W[N ], (58)

we have, for i = 1, 2,

f2i−1(U1)− f2i−1(U2) = −

(1

(nil + nδi1)2S[nil + nδi1, ji1, θ + θδi1](nδi1x)2

− 1

(nil + nδi2)2S[nil + nδi2, ji2, θ + θδi2](nδi2x)2

)

+

(1

nil + nδi1

(θδi1xn

δi1x +

2j2i1

(nil + nδi1)3(nδi1x)2

)

A FULL BIPOLAR HD MODEL FOR SEMICONDUCTORS 1611

− 1

nil + nδi2

(θδi2xn

δi2x +

2j2i2

(nil + nδi2)3(nδi2x)2

))

+

(1

nil

(θδi1n

δi1xx −

j2i1

(nil + nδi1)2nδi1xx

)− 1

nil

(θδi2n

δi2xx −

j2i2

(nil + nδi2)2nδi2xx

))

+

(( 1

nil + nδi1− 1

nil

)S[nil + nδi1, ji1, θ + θδi1]nδi1xx

−( 1

nil + nδi2− 1

nil

)S[nil + nδi2, ji2, θ + θδi2]nδi2xx

)

+

(3

2ji1θ

δi1x −

3

2ji2θ

δi2x

)−

(θ + θδi1

2(nil + nδi1)ji1n

δi1x −

θ + θδi22(nil + nδi2)

ji2nδi2x

)

−

(j2i1

2(nil + nδi1)− j2

i2

2(nil + nδi2)

)+

(3

2nδi1θ

δi1 −

3

2nδi2θ

δi2

), (59)

f2i(U1)− f2i(U2) = −

(ji1θ

δi1x − ji2θδi2x

)

+

(θ + θδi1

3(nil + nδi1)ji1n

δi1x −

θ + θδi23(nil + nδi2)

ji2nδi2x

)

+

(j2i1

3(nil + nδi1)− j2

i2

3(nil + nδi2)

)−

(nδi1θ

δi1 − nδi2θδi2

), (60)

By the similar way in [25], we obtain

|ji1 − ji2| ≤ C(nil)(‖nδi1 − nδi2‖1 + ‖(θδi1 − θδi2)x‖). (61)

Through the straightforward computations, for i = 1, 2, we have

S[nil + nδi1, ji1, θ + θδi1]− S[nil + nδi2, ji2, θ + θδi2]

=

((θ + θδi1)− j2

i1

(nil + nδi1)2

)−

((θ + θδi2)− j2

i2

(nil + nδi2)2

)

=(θδ1 − θδ2)−

(1

(nil + nδi1)2− 1

(nil + nδi2)2

)j2i1 −

1

(nil + nδi2)2(j2i1 − j2

i2)

=(θδ1 − θδ2) +2j2i1

((nil + nδi2) + α(nδi1 − nδi2))3(nδ1 − nδ2)

− 1

(nil + nδi2)2(ji1 + ji2)(ji1 − ji2), (62)

where 0 < α < 1. From the Sobolev embedding theorem, (49) and (54), we have

1612 HAIFENG HU AND KAIJUN ZHANG

1

2nil ≤ nil + nδik ≤ 2nil,

1

2θ ≤ θ + θδik ≤ 2θ,

1

2θ ≤ S[nil + nδik, jik, θ + θδik] ≤ 2θ, i, k = 1, 2, (63)

provided 0 < δ 1. Applying the similar method in (62) to the each term in (59)and (60), and then taking the L2-norm, we can see that the estimates

‖f2i−1(U1)− f2i−1(U2)‖ ≤ C(nil, θ, N)δ(‖nδi1 − nδi2‖2 + ‖θδi1 − θδi2‖1), (64)

and

‖f2i(U1)− f2i(U2)‖ ≤ C(nil, θ, N)δ(‖nδi1 − nδi2‖1 + ‖θδi1 − θδi2‖1), (65)

follow from (49), (61)-(63) and Taylor’s formula.Differentiating (59) and (60) with respect to x and taking the L2-norm, by the

similar way in (64) and (65), we obtain the estimates

‖∂x(f2i−1(U1)− f2i−1(U2))‖ ≤ C(nil, θ, N)δ(‖nδi1 − nδi2‖3 + ‖θδi1 − θδi2‖2), (66)

and

‖∂x(f2i(U1)− f2i(U2))‖ ≤ C(nil, θ, N)δ(‖nδi1 − nδi2‖2 + ‖θδi1 − θδi2‖2), (67)

Synthesizing the estimates (64), (65), (66) and (67), we have

‖f(U1)− f(U2)‖1 ≤ C(n1l, n2l, θ, N)δ‖U1 − U2‖3. (68)

Substituting (54) and (68) into (57), we obtain

‖S1U1 − S1U2‖3 ≤ C6(n1l, n2l, θ)δ‖U1 − U2‖3, ∀U1, U2 ∈W[N ]. (69)

From (69) we know, if 0 < δ < C−16 , then S1 is a contraction mapping.

From (55), (69) and the Banach fixed-point theorem, the mapping S1 has aunique fixed-point U ∈ W[N ] provided 0 < δ 1. Finally, the unique solution

(n1, θ1, n2, θ2) ∈ [B2(Ω)]4 of (24)-(25) is constructed from the unique fixed-point by(20). In addition, the estimate (26) is valid by the Sobolev embedding theorem.

Proof of Theorem 1.1. Based on the result in Lemma 2.1,we can use the unique solu-tion (n1, θ1, n2, θ2) of the BVP (24)-(25) to construct the unique solution (n1, j1, θ1,

n2, j2, θ2, φ) ∈ [B2(Ω)]6 × B4(Ω) of the stationary problem (13)-(16). In fact, the

only thing we need to do is to define a constant ji := Ji[ni, θi], i = 1, 2 and a func-

tion φ := Φ[n1, n2], where Ji and Φ are given in (23) and (21). In addition, thisstationary solution satisfies the conditions (7), thanks to (63). Finally, the estimate(11) follows from (26), (21) and (49).

3. Asymptotic stability of the stationary solution.

3.1. Local existence. In this subsection, we discuss the existence and uniquenessof the time local solution of (1)-(5). It can be established using the same argumentsin [25]. Here we only state the main results and omit the details (see [25]).

For smooth solution, the full bipolar HD model (1) is equivalent to the followingquasi-linear system

A FULL BIPOLAR HD MODEL FOR SEMICONDUCTORS 1613

nit + jix = 0, (70a)

jit +(θi −

j2i

n2i

)nix + 2

jinijix + niθix = (−1)i−1niφx − ji, (70b)

niθit + jiθix +2

3

( jini

)xniθi −

2

3θixx =

1

3

j2i

ni− ni(θi − θ), (70c)

φxx = n1 − n2 −D(x), ∀(t, x) ∈ R+ × Ω. (70d)

Similar to (21), we have the formula for the time-dependent electrostatic potentialφ(t, x) as follows

φ(t, x) = Φ[n1, n2](t, x)

:=

∫ x

0

∫ y

0

n1(t, z)− n2(t, z)−D(z)dzdy

+

(φr −

∫ 1

0

∫ y

0

n1(t, z)− n2(t, z)−D(z)dzdy

)x. (71)

In order to clarify the idea of the proof, we write the system (70) as the formL(U)U = F . Formally, a solution of the quasi-linear system L(U)U = F willbe obtained by finding a fixed-point of the map V 7→ U given by L(V )U = F ,so we need an existence theorem for the linearized problem. To this end, for any

0 < T < +∞, we study the linearized problem for the unknown (n1, j1, θ1, n2, j2, θ2)on [0, T ],

nit + jix = 0, (72a)

jit +(θi −

j2i

n2i

)nix + 2

jinijix + niθix = (−1)i−1niφx − ji, (72b)

niθit + jiθix +2

3

( jini

)xniθi −

2

3θixx =

1

3

j2i

ni− ni(θi − θ), (72c)

φ = Φ[n1, n2], ∀(t, x) ∈ QT := [0, T ]× Ω, (72d)

with the initial data (2) and the boundary data (3)-(4). Here Φ in (72d) is givenby (71). Before stating the main results, we firstly introduce the solution space asfollows

X(T ;m,M) :=

(n1, j1, θ1, n2, j2, θ2)(t, x)∣∣∣ (ni, ji, θi)(0, x) = (ni0, ji0, θi0)(x),

(ni, ji, θi) ∈ S[QT ], ni(t, x), θi(t, x), S[ni, ji, θi](t, x) ≥ m,∀(t, x) ∈ QT ,

i = 1, 2,

2∑i=1

‖(ni, ji, θi)‖S[QT ] ≤M, (73)

where S[QT ] is a Banach space defined as

S[QT ] := X2([0, T ])× [X11([0, T ]) ∩H2(0, T ;L2(Ω))]

× [Y([0, T ]) ∩H1(0, T ;H1(Ω))], (74)

endowed with the norm

‖(n, j, θ)‖S[QT ] :=‖n‖X2([0,T ]) + ‖j‖X11([0,T ]) + ‖θ‖Y([0,T ]) + ‖(θxt, jtt)‖L2(QT )

1614 HAIFENG HU AND KAIJUN ZHANG

= supt∈[0,T ]

(‖(n, j, θ)(t)‖2 + ‖(nt, jt)(t)‖1 + ‖(ntt, θt)(t)‖

)+ ‖(θxt, jtt)‖L2(QT ). (75)

Firstly, for any constants 0 < T <∞, m > 0 and M > 0, it is obvious that the setX(T ;m,M) is a closed subset of the product space [S[QT ]]2. In addition, for anyinitial data (n10, j10, θ10, n20, j20, θ20) ∈ H2(Ω) satisfying (7), we define

m0 := mini=1,2

infx∈Ωni0(x), θi0(x), S[ni0, ji0, θi0](x) > 0, (76a)

M0 :=

2∑i=1

‖(ni0, ji0, θi0)‖2 > 0, (76b)

one can see that X(T ;m,M) 6= ∅ provided 0 < m ≤ m0 and M ≥ M0.Now, applying the Galerkin method (cf. [27, 30]) and the theory of symmetric

hyperbolic system (cf. Theorem-A1 in [26]) we can obtain the existence theorem ofthe linearized problem (72) and (2)-(4) as follows.

Lemma 3.1. Under the assumptions of Theorem 1.2. Let T > 0, 0 < m ≤m0 and M ≥ M0 are arbitrary constants and D ∈ B0(Ω), if the coefficients(n1, j1, θ1, n2, j2, θ2) ∈ X(T ;m,M), then the linearized problem (72) and (2)-(4)

has a unique solution (n1, j1, θ1, n2, j2, θ2) ∈ [S[QT ]]2. Moreover, if we denote the

solution operator as S2 : (n1, j1, θ1, n2, j2, θ2) 7→ (n1, j1, θ1, n2, j2, θ2), then thereexist certain positive constants T0, m0(≤ m0) and M0(≥ M0) depend on the initialdata such that X(T0;m0,M0) is invariant under the operator S2.

Next, using the Lemma 3.1, the standard iterative arguments and the energymethod, we can establish the existence theorem of the original problem (1)-(5) asfollows.

Lemma 3.2 (Local existence). Under the assumptions of Theorem 1.2. Let D ∈B0(Ω), then there exists a constant T1 > 0 depends on the initial data such thatthe (1)-(5) has a unique solution (n1, j1, θ1, n2, j2, θ2, φ) ∈ [S[QT1

]]2 × X22([0, T1]).

Moreover, the solution (n1, j1, θ1, n2, j2, θ2, φ) verifies the condition (7).

3.2. A priori estimates. We regard the solution (n1, j1, θ1, n2, j2, θ2, φ) as a per-

turbation from the stationary solution (n1, j1, θ1, n2, j2, θ2, φ) in showing its stabil-ity.

ψi(t, x) := ni(t, x)− ni(x), ηi(t, x) := ji(t, x)− ji,

χi(t, x) := θi(t, x)− θi(x), σ(t, x) := φ(t, x)− φ(x). (77)

From (1b)/ni and (1a), we have( jini

)t

+1

2

( j2i

n2i

)x

+ θi(lnni)x + θix = (−1)i−1φx −jini. (78)

Similarly, we have the stationary version of the equation (78) as follows

1

2

( j2i

n2i

)x

+ θi(ln ni)x + θix = (−1)i−1φx −jini. (79)

A FULL BIPOLAR HD MODEL FOR SEMICONDUCTORS 1615

From (1a)− (13a), (1c)− (13c), (1d)− (13d) and (78)− (79), we get the system forthe perturbation (ψ1, η1, χ1, ψ2, η2, χ2, σ) as follows

ψit + ηix = 0, (80a)( ji + ηini + ψi

)t

+1

2

( (ji + ηi)2

(ni + ψi)2− j2

i

n2i

)x

+ θi(ln(ni + ψi)− ln ni)x

+(ni + ψi)xni + ψi

χi + χix + (−1)iσx +ji + ηini + ψi

− jini

= 0,

(80b)

(ni + ψi)χit +2

3(θi + χi)ηix −

2

3(ji + ηi)

θi + χini + ψi

ψix −2

3χixx = Gi, (80c)

σxx = ψ1 − ψ2, (80d)

where

Gi :=− jiχix − θixηi +2nixθi

3niηi +

2jinix3ni

χi −2jiθinix3nini

ψi

+2ji + ηi

3niηi −

j2i ψi

3nini− (θi − θ)ψi − niχi. (81)

The initial and boundary data posed on the system (80) are derived from (2)-(5)as follows

ψi(0, x) = ψi0(x) := ni0(x)− ni(x), (82a)

ηi(0, x) = ηi0(x) := ji0(x)− ji, (82b)

χi(0, x) = χi0(x) := θi0(x)− θi(x), (82c)

and

ψi(t, 0) = ψi(t, 1) = 0, (83a)

χi(t, 0) = χi(t, 1) = 0, (83b)

σ(t, 0) = σ(t, 1) = 0. (83c)

Theorem 1.1 and Lemma 3.2 guarantee the existence of the time local solution(ψ1, η1, χ1, ψ2, η2, χ2, σ) of the IBVP (80)-(83). It is summarized in the followingcorollary.

Corollary 1. Suppose that the initial data (ψ10, η10, χ10, ψ20, η20, χ20) ∈ H2(Ω) and

(n1+ψ10, j1+η10, θ1+χ10, n2+ψ20, j2+η20, θ2+χ20) satisfies (7). Then there existsa constant T2 > 0 such that (80)-(83) has a unique solution (ψ1, η1, χ1, ψ2, η2, χ2, σ)∈ [S[QT2

]]2 × X22([0, T2]) with the subsonic property (7).

To show the existence of the global solution, it is sufficient to derive an a prioriestimate for a local solution given by Corollary 1. For convenience, we introducethe notation

N(T ) := supt∈[0,T ]

2∑i=1

‖(ψi, ηi, χi)(t)‖2. (84)

The main aim in Sect.3 is to show the following Theorem.

Theorem 3.3 (Uniform a priori estimate). Let (ψ1, η1, χ1, ψ2, η2, χ2, σ) ∈ [S[QT ]]2

×X22([0, T ]) be a local solution on a finite time interval [0, T ] of the IBVP (80)-(83).

1616 HAIFENG HU AND KAIJUN ZHANG

Then there exist the positive constants ε0, C and γ such that if N(T )+δ ≤ ε0, thenit holds that

2∑i=1

‖(ψi, ηi, χi)(t)‖22 + ‖σ(t)‖24 ≤ C2∑i=1

‖(ψi, ηi, χi)(0)‖22 e−γt, ∀t ∈ [0, T ], (85)

where the positive constants C and γ are independent of time T > 0.

The proof of Theorem 3.3 is based on several steps of technical energy estimateswhich are stated as a series of lemmas in Sect.3.3 and Sect.3.4.

3.3. Basic estimate. In this subsection, we derive the basic energy estimate. Tothis end, we employ an energy form E defined by

E(t, x) :=1

2σ2x +

2∑i=1

( η2i

2ni+ niθiΨ

( nini

)+

3

2niθiΨ

(θiθi

)), (86)

whereΨ(s) := s− 1− ln s, ∀s > 0. (87)

We can easily see that Ψ(s) is equivalent to |s − 1|2 on any sub-interval [s0,∞) ⊂(0,∞).

Lemma 3.4 (Elliptic estimates). Suppose that the assumption in Theorem 3.3holds. Then the following estimates hold for ∀t ∈ [0, T ],

‖∂kt σ(t)‖2 ≤ C‖(∂kt ψ1, ∂kt ψ2)(t)‖, k = 0, 1, 2, (88)

‖σxt(t)‖ ≤ C‖(η1, η2)(t)‖, (89)

2∑i=1

|(ψit, ηit)(t)|0 ≤ CN(T ), (90)

where C is a positive constant independent of time T .

Proof. From (80d) and (83c), we have

σ(t, x) =

∫ x

0

∫ y

0

(ψ1 − ψ2)(t, z)dzdy − x∫ 1

0

∫ y

0

(ψ1 − ψ2)(t, z)dzdy. (91)

By virtue of (91) and (80a), the estimates (88) and (89) immediately hold. Solvingψit and ηit out from (80a) and (80b), respectively. Applying the Sobolev embeddingtheorem to (84), and using (88) and (11), we get (90).

Lemma 3.5 (Basic estimate). Suppose that the assumption in Theorem 3.3 holds.Then the following differential identity holds for all t ∈ [0, T ],

d

dt

∫ 1

0

E(t, x)dx+

∫ 1

0

2∑i=1

(η2i

ni+

3ni2θi

χ2i +

χ2ix

θi

)dx =

∫ 1

0

(R2 +R4)dx, (92)

where R2 and R4 are defined in (102) and (105), respectively. Moreover, if 0 <N(T ) + δ 1, then we get∣∣∣ ∫ 1

0

(R2 +R4)dx∣∣∣ ≤ C(N(T ) + δ)

2∑i=1

‖(ψi, ηi, χi)(t)‖21, (93)

C7

2∑i=1

‖(ψi, ηi, χi)(t)‖2 ≤∫ 1

0

E(t, x)dx ≤ C8

2∑i=1

‖(ψi, ηi, χi)(t)‖2, (94)

where the positive constants C, C7 and C8 are independent of T .

A FULL BIPOLAR HD MODEL FOR SEMICONDUCTORS 1617

Proof. Multiplying (80b) with ηi leads to( jini

)tηi +

1

2

( j2i

n2i

− j2i

n2i

)xηi + θi(lnni − ln ni)xηi

+nixniχiηi + χixηi + (−1)iσxηi +

( jini− jini

)ηi = 0. (95)

In the bipolar case, we have to treat the sixth term in the left hand side of (95) asfollows

(−1)iσxηi = (−1)i((σηi)x − σηix) = (−1)i(σηi)x + (−1)i+1σηix, (96)

and the other terms can be treated with the similar method in [25]. Thus we canrewrite (95) as( η2

i

2ni

)t

+ηi + 2ji

2n2i

ηixηi +1

2

( j2i

n2i

− j2i

n2i

)xηi +

(niθiΨ

( nini

))t

+(θi(lnni − ln ni)ηi

)x− θix(lnni − ln ni)ηi +

nixniχiηi +

ψixni

χiηi + χixηi

+ (−1)i(σηi)x + (−1)i+1σηix + ji

( 1

ni− 1

ni

)ηi +

1

niη2i = 0, i = 1, 2. (97)

Summing (97) over i, we obtain

− (ση1)x + ση1x + (ση2)x − ση2x

+

2∑i=1

( η2i

2ni+ niθiΨ

( nini

))t

+1

niη2i +

nixniχiηi + χixηi

+(θi(lnni − ln ni)ηi

)x

+ηi + 2ji

2n2i

ηixηi +1

2

( j2i

n2i

− j2i

n2i

)xηi

+ ji

( 1

ni− 1

ni

)ηi − θix(lnni − ln ni)ηi +

ψixni

χiηi

= 0. (98)

We calculate the first four terms in the l.h.s. of (98) as follows

the first four terms =− (σ(η1 − η2))x + σ(η1x − η2x)

=− (σ(η1 − η2))x − σ(ψ1 − ψ2)t

=− (σ(η1 − η2))x − σσxxt=− (σ(η1 − η2))x − ((σσxt)x − σxσxt)

=− (σ(η1 − η2))x − (σσxt)x +(1

2σ2x

)t

=− (σσxt + σ(η1 − η2))x +(1

2σ2x

)t. (99)

Substituting (99) into (98), we have1

2σ2x+

2∑i=1

( η2i

2ni+niθiΨ

( nini

))t+

2∑i=1

( 1

niη2i +

nixniχiηi+χixηi

)= R1x+R2, (100)

where

R1 := σσxt + σ(η1 − η2)−2∑i=1

θi(lnni − ln ni)ηi, (101)

1618 HAIFENG HU AND KAIJUN ZHANG

R2 := −2∑i=1

ηi + 2ji2n2

i

ηixηi +1

2

( j2i

n2i

− j2i

n2i

)xηi

+ ji

( 1

ni− 1

ni

)ηi − θix(lnni − ln ni)ηi +

ψixni

χiηi

. (102)

From (80c)×(3χi/2θi), and summing the resultants over i from 1 to 2, we obtain

( 2∑i=1

3

2niθiΨ

(θiθi

))t

+

2∑i=1

(3ni2θi

χ2i +

χ2ix

θi− nix

niχiηi − χixηi

)= R3x +R4, (103)

where

R3 :=

2∑i=1

(− ηiχi +

χiχixθi

), (104)

R4 :=

2∑i=1

− 3

2θiηixΨ

(θiθi

)− 3

2

(jiχix + θixηi −

2ji + ηi3ni

ηi +j2i ψi

3nini+ (θi − θ)ψi

)χiθi

+(jiθiniψix +

jinixni

χi −jiθinixnini

ψi +θixθiχix

)χiθi

. (105)

From (100)+(103), we have

Et +

2∑i=1

(η2i

ni+

3ni2θi

χ2i +

χ2ix

θi

)= (R1 +R3)x +R2 +R4, (106)

where the terms nix

niχiηi + χixηi appear in both (100) and (103), and cancel each

other. Integrating (106) over Ω immediately yields (92), where we have used thefact ∫ 1

0

(R1 +R3)xdx = 0, (107)

which follows from the homogeneous boundary conditions (83). From (11), theSobolev embedding theorem and Cauchy-Schwartz inequality, we get (93). If 0 <N(T ) + δ 1, then the equivalence of the function Ψ(s) implies (94).

Remark 1. One can see that we lost the desired zero order term like∫ 1

0O+(1)(ψ2

1 +

ψ22)dx for the perturbation densities ψi in the integral term of the left hand side

of the basic estimate (92), where the notation O+(1) means some positive bounded

quantity. The reason is that the procedure∑2i=1

∫ 1

0(80b) × (−1)iσxdx (i.e. the

estimate for the perturbation electric field σx) yields the term like∫ 1

0O+(1)(ψ1 −

ψ2)2dx rather than the desired one. Thus, we meet with a new difficulty on the wayto close the uniform estimate (85). It is the essential difference between unipolarcase [25] and our bipolar case. Fortunately, this new difficulty can be overcomein the higher order estimates by using the Poincare inequality and the smallnessestimates (11b) of the stationary solution (see (153)).

A FULL BIPOLAR HD MODEL FOR SEMICONDUCTORS 1619

3.4. Higher order estimates. This subsection is devoted to the derivation of thehigher order estimates. On the one hand, in consideration of the homogeneousboundary conditions (83), we can only derive the estimates of the time derivatives.This will be reflected in the choice of the multipliers later, namely, all of the mul-tipliers ∂kt ψi, ∂

kt ψit and χit are time derivatives. On the other hand, because time

derivatives can control the spatial derivatives (see (122)), deriving only the esti-mates of the time derivatives is enough. To justify these computations, we needthe arguments using the mollifier with respect to the time variable t since the reg-ularity of the solution (ψ1, η1, χ1, ψ2, η2, χ2) is not enough. However, we omit thesearguments as they are standard.

For convenience, we introduce the notations as follows

A2−1(t) :=

2∑i=1

‖(ψi, ηi, χi, χix)(t)‖2, (108)

A2k(t) := A2

−1(t) +

k∑l=0

2∑i=1

‖(∂ltψit, ∂ltψix)(t)‖2, k = 0, 1. (109)

Next, we derive the working equations which are used to obtain the higher orderestimates. From −∂kt ∂x(1b)/ni − ∂x(13b)/ni with k = 0, 1, we get one workingequation

n−1i ∂kt ψitt − ((θin

−1i − j

2i n−3i )∂kt ψix)x + (−1)i+1∂kt (ψ1 − ψ2)

+ n−1i ∂kt ψit − ∂kt χixx = −2jin

−2i ∂kt ψixt + ∂kt Fi +Kik, (110)

where

Fi := 2ψ2itn−2i + 4jin

−3i nixψit + 2j2

i n−4i (2nix + ψix)ψix

+ 2θixn−1i ψix + (j2

i n−3i )xψix − θixn−1

i ψix + θin−2i nixψix + 2j2

i n2ix(n−4

i − n−4i )

+ 2(2ji + ηi)n−4i n2

ixηi − j2i nixx(n−3

i − n−3i )− (2ji + ηi)n

−3i nixxηi

− 2nixθix(nini)−1ψi − θinixx(nini)

−1ψi + nixxn−1i χi + 2nixn

−1i χix

+ (−1)i+1(φxnix(nini)

−1ψi − nixn−1i σx − φxn−1

i ψix

), (111)

Ki0 := 0, Ki1 := n−2i ψitψitt + ((θin

−1i − j

2i n−3i )tψix)x

+ n−2i ψ2

it − 2(jin−2i )tψixt. (112)

From ∂kt (80c) with k = 0, 1, we have the other working equation

ni∂kt χit +

2

3θi∂

kt ηix −

2

3jiθin

−1i ∂kt ψix −

2

3∂kt χixx = ∂kt Gi + Lik, (113)

where

Li0 := 0, Li1 := −ψitχit −2

3ηixχit +

2

3(jiθin

−1i )tψix. (114)

From (111), (112), (84), (11) and Sobolev embedding theorem, we have theestimates

‖Fi‖ ≤ C(N(T ) + δ)‖(ψit, ψix, ηi, χi, χix, ψ1, ψ2)‖, (115)

‖Ki1‖+ ‖Fit‖ ≤ C(N(T ) + δ)‖(ψitt, ψixt, ηit, ηixt, χit, χixt, ψ1t, ψ2t)‖. (116)

1620 HAIFENG HU AND KAIJUN ZHANG

By the similar way, from (81) and (114), we get the estimates

‖Gi‖ ≤ C(N(T ) + δ)‖(ψi, ηi, χix)‖+ C‖χi‖, (117)

‖Li1‖+ ‖Git‖ ≤ C(N(T ) + δ)‖(ψit, ηit, χixt)‖+ C‖χit‖. (118)

Moreover, the estimates

‖∂kt ηix(t)‖ = ‖∂kt ψit(t)‖, ‖ηixx(t)‖ = ‖ψixt(t)‖, (119)

‖σ(t)‖4 ≤ C‖(ψ1, ψ2)(t)‖2, (120)

‖∂kt ηit(t)‖ ≤ C(Ak(t) + k‖(χit, χixt)(t)‖), (121)

C9

2∑i=1

‖(ψi, ηi, χi)(t)‖2 ≤ A1(t) + ‖(χ1t, χ2t)(t)‖ ≤ C10

2∑i=1

‖(ψi, ηi, χi)(t)‖2,

(122)

immediately follow from (80). The estimate (122) means that we achieve the objec-tive that the time derivatives control the spatial derivatives. Now it only remainsto estimate the time derivatives through the working equations (110) and (113) asfollows.

Lemma 3.6. Suppose that the assumption in Theorem 3.3 holds. Then the follow-ing differential identity holds for all t ∈ [0, T ] and k = 0, 1,

d

dtI

(k)1 (t) +

∫ 1

0

2∑i=1

(θin−1i (∂kt ψix)2 − n−1

i (∂kt ψit)2)

+ (∂kt ψ1 − ∂kt ψ2)2dx = J

(k)1 (t), (123)

where I(k)1 (t) and J

(k)1 (t) are defined in (130). Moreover, if 0 < N(T )+δ 1, then

we have|I(k)

1 (t)| ≤ CA2k(t), (124)

|J (k)1 (t)| ≤ µ

2∑i=1

‖∂kt ψi(t)‖21 + Cµ(N(T ) + δ)(A2k(t)

+ k

2∑i=1

‖χit(t)‖2)

+ Cµ

2∑i=1

‖∂kt χix(t)‖2, (125)

where the positive constants C, µ and Cµ are independent of T , and µ will bedetermined later.

Proof. From (83), we get

∂kt ψi(t, 0) = ∂kt ψi(t, 1) = 0, ∀t ∈ [0, T ], i = 1, 2, k = 0, 1. (126)

By the procedure∫ 1

0(110)× ∂kt ψidx, we have

d

dtI

(k)i1 (t) +

∫ 1

0

(θin−1i (∂kt ψix)2

+ (−1)i+1(∂kt ψ1 − ∂kt ψ2)∂kt ψi − n−1i (∂kt ψit)

2)dx = J

(k)i1 (t), (127)

where

I(k)i1 (t) :=

∫ 1

0

(n−1i ∂kt ψit∂

kt ψi + (2ni)

−1(∂kt ψi)2)dx, (128)

A FULL BIPOLAR HD MODEL FOR SEMICONDUCTORS 1621

J(k)i1 (t) :=

∫ 1

0

((n−1i )t∂

kt ψi∂

kt ψit + (2jin

−2i ∂kt ψi)x∂

kt ψit + j2

i n−3i (∂kt ψix)2

− ∂kt χix∂kt ψix + ((2ni)−1)t(∂

kt ψi)

2 + (∂kt Fi +Kik)∂kt ψi

)dx. (129)

Summing (127) over i from 1 to 2, we obtain (123). Here we need to define

I(k)1 (t) :=

2∑i=1

I(k)i1 (t), J

(k)1 (t) :=

2∑i=1

J(k)i1 (t), (130)

and have used the fact∫ 1

0

2∑i=1

(−1)i+1(∂kt ψ1 − ∂kt ψ2)∂kt ψidx =

∫ 1

0

(∂kt ψ1 − ∂kt ψ2)2dx. (131)

By using the Cauchy-Schwartz inequality, (116), (115) and Lemma3.4, we can di-rectly estimate (130). After a tedious and laborious calculation, we obtain (124)and (125).

Lemma 3.7. Suppose that the assumption in Theorem 3.3 holds. Then the follow-ing differential identity holds for all t ∈ [0, T ] and k = 0, 1,

d

dtI

(k)2 (t) +

∫ 1

0

2∑i=1

(n−1i + θi)(∂

kt ψit)

2dx = J(k)2 (t), (132)

where I(k)2 (t) and J

(k)2 (t) are defined in (140). Moreover, if 0 < N(T )+δ 1, then

there are the estimates

|I(k)2 (t)| ≤ C(A2

k(t) +

2∑i=1

‖∂kt χi(t)‖2), (133)

|J (k)2 (t)| ≤ (Cµ(N(T ) + δ) + µ)A2

k(t)

+ Cµ(N(T ) + δ)

2∑i=1

‖χit(t)‖2k + Cµ

2∑i=1

‖∂kt χi(t)‖21, (134)

where the positive constants C, µ and Cµ are independent of time T , and µ will bedetermined later.

Proof. From (83), we have

∂kt ψit(t, 0) = ∂kt ψit(t, 1) = 0, ∀t ∈ [0, T ], i = 1, 2, k = 0, 1. (135)

It follows from the procedure∫ 1

0(110)× ∂kt ψitdx that

d

dtI

(k)i2 (t) +

∫ 1

0

(−1)i+1(∂kt ψ1 − ∂kt ψ2)∂kt ψitdx

+

∫ 1

0

(n−1i + θi)(∂

kt ψit)

2dx = J(k)i2 (t), (136)

where

I(k)i2 (t) :=

∫ 1

0

(1

2n−1i (∂kt ψit)

2 +1

2(θin

−1i − j

2i n−3i )(∂kt ψix)2

+3

2ni∂

kt χi∂

kt ηix

)dx, (137)

1622 HAIFENG HU AND KAIJUN ZHANG

J(k)i2 (t) :=

∫ 1

0

(((2ni)

−1)t(∂kt ψit)

2

+1

2(θin

−1i − j

2i n−3i )t(∂

kt ψix)2 +

3

2ψit∂

kt χi∂

kt ηix + (jin

−2i )x(∂kt ψit)

2

− jiθin−1i ∂kt ψix∂

kt ψit −

3

2(∂kt Gi + Lik)∂kt ψit −

3

2(ni∂

kt χi)x∂

kt ηit

+ (∂kt Fi +Kik)∂kt ψit

)dx. (138)

In order to obtain (132), we sum (136) over i from 1 to 2, and then we find that thefirst integral term in the left hand side of the resultant summation can be calculatedas follows

2∑i=1

∫ 1

0

(−1)i+1(∂kt ψ1 − ∂kt ψ2)∂kt ψitdx

=

∫ 1

0

((∂kt ψ1 − ∂kt ψ2)∂kt ψ1t − (∂kt ψ1 − ∂kt ψ2)∂kt ψ2t

)dx

=

∫ 1

0

(∂kt ψ1(∂kt ψ1)t − ((∂kt ψ1)t∂

kt ψ2 + ∂kt ψ1(∂kt ψ2)t) + ∂kt ψ2(∂kt ψ2)t

)dx

=

∫ 1

0

(1

2((∂kt ψ1)2)t − (∂kt ψ1∂

kt ψ2)t +

1

2((∂kt ψ2)2)t

)dx

=d

dt

∫ 1

0

1

2(∂kt ψ1 − ∂kt ψ2)2dx. (139)

Substituting (139) into the resultant summation∑2i=1 (136), we obtain (132) by

using the definition

I(k)2 (t) :=

∫ 1

0

1

2(∂kt ψ1 − ∂kt ψ2)2dx+

2∑i=1

I(k)i2 (t), J

(k)2 (t) :=

2∑i=1

J(k)i2 (t). (140)

Next, in view of (117) and (118), we obtain (133) and (134) by the similar wayin Lemma 3.6.

In order to complete the proof of the uniform estimate (85), we derive the higherorder estimates of χi.

Lemma 3.8. Suppose that the assumption in Theorem 3.3 holds. Then the follow-ing differential identity holds for all t ∈ [0, T ],

d

dtI3(t) +

∫ 1

0

2∑i=1

niχ2itdx = J3(t), (141)

where I3(t) and J3(t) are defined as follows

I3(t) :=

2∑i=1

∫ 1

0

(1

3χ2ix +

2

3θiηixχi

)dx, (142a)

J3(t) :=

2∑i=1

∫ 1

0

(2

3χitηixχi −

2

3(θiχi)xηit +

(2

3jiθin

−1i ψix +Gi

)χit

)dx. (142b)

Moreover, if 0 < N(T ) + δ 1, then there are the estimates

|I3(t)| ≤ CA20(t), (143)

A FULL BIPOLAR HD MODEL FOR SEMICONDUCTORS 1623

|J3(t)| ≤ (Cµ(N(T ) + δ) + µ)A20(t)

+ (C(N(T ) + δ) + µ)

2∑i=1

‖χit(t)‖2 + Cµ

2∑i=1

‖χi(t)‖21, (144)

where the positive constants C, µ and Cµ are independent of time T , and µ will bedetermined later.

Proof. From the procedure∫ 1

0(113)|k=0 × χitdx, we can prove this Lemma be-

cause the estimates for χi has the same mathematical structure comparing with theunipolar case [25] . Here we omit the details.

Lemma 3.9. Suppose that the assumption in Theorem 3.3 holds. Then the follow-ing differential identity holds for all t ∈ [0, T ],

d

dtI4(t) +

∫ 1

0

2∑i=1

2

3χ2ixtdx = J4(t), (145)

where I4(t) and J4(t) are defined as follows

I4(t) :=

2∑i=1

∫ 1

0

1

2niχ

2itdx, (146a)

J4(t) :=

2∑i=1

∫ 1

0

(1

2ψitχ

2it +

2

3(θiχit)xηit

− 2

3(jiθin

−1i χit)xψit + (Git + Li1)χit

)dx. (146b)

Moreover, if 0 < N(T ) + δ 1, then there are the estimates

|I4(t)| ≤ C2∑i=1

‖χit(t)‖2, (147)

|J4(t)| ≤ Cµ(

2∑i=1

‖χit(t)‖2 +A20(t)) + (C(N(T ) + δ) + µ)

2∑i=1

‖χixt(t)‖2, (148)

where the positive constants C, µ and Cµ are independent of time T , and µ will bedetermined later.

Proof. From the procedure∫ 1

0(113)|k=1 × χitdx, we can complete the proof due to

the same reason in Lemma 3.8.

Proof of Theorem 3.3. From the following procedure

(92) + β(

(132)|k=0 + (141))

+ αβ(123)|k=0

+ β2(145) + β3(132)|k=1 + αβ3(123)|k=1, (149)

where α and β are positive constants determined later, we obtain the ultimatedifferential identity

d

dtE(t) + F (t) = 0, ∀t ∈ [0, T ], (150)

1624 HAIFENG HU AND KAIJUN ZHANG

where

E(t) :=

∫ 1

0

E(t, x)dx+ β(I

(0)2 (t) + I3(t)

)+ αβI

(0)1 (t) + β2I4(t) + β3I

(1)2 (t) + αβ3I

(1)1 (t), (151)

F (t) :=

∫ 1

0

2∑i=1

(η2i

ni+

3ni2θi

χ2i +

χ2ix

θi

)dx−

∫ 1

0

(R2 +R4)dx

+ β

∫ 1

0

2∑i=1

(n−1i + θi)ψ

2itdx− βJ

(0)2 (t) + β

∫ 1

0

2∑i=1

niχ2itdx− βJ3(t)

+ αβ

∫ 1

0

2∑i=1

(θin−1i ψ2

ix − n−1i ψ2

it

)+ (ψ1 − ψ2)2

dx− αβJ (0)

1 (t)

+ β2

∫ 1

0

2∑i=1

2

3χ2ixtdx− β2J4(t) + β3

∫ 1

0

2∑i=1

(n−1i + θi)(ψitt)

2dx− β3J(1)2 (t)

+ αβ3

∫ 1

0

2∑i=1

(θin−1i ψ2

ixt − n−1i ψ2

itt

)+ (ψ1t − ψ2t)

2dx− αβ3J

(1)1 (t). (152)

Taking α, µ, β and N(T ) + δ sufficiently small in the order: 0 < N(T ) + δ β3 β2 β µ α 1. From the estimates (94), (124), (133), (143) and

(147), we can prove that the quantity E(t) is equivalent to A21(t) + ‖(χ1t, χ2t)(t)‖2.

Similarly, from the estimates (93) (125), (134), (144) and (148), we can also show

that the quantity F (t) can be controlled by A21(t) + ‖(χ1t, χ2t)(t)‖2 from below.

In estimating the lower-bound of F (t), in order to compensate the loss of the“nice” term in the basic estimate just as mentioned in Remark 1, we have appliedthe Poincare inequality to the first term of the integral in the left hand side of(123)|k=0 as follows∫ 1

0

2∑i=1

θin−1i ψ2

ixdx ≥ C11

∫ 1

0

(ψ21x + ψ2

2x)dx

=C11

2

∫ 1

0

(ψ21x + ψ2

2x)dx+C11

2

∫ 1

0

(ψ21x + ψ2

2x)dx

≥ C12

2

∫ 1

0

(ψ21 + ψ2

2)dx+C11

2

∫ 1

0

(ψ21x + ψ2

2x)dx

≥ C13

∫ 1

0

2∑i=1

(ψ2i + ψ2

ix)dx, (153)

where C11, C12 and C13 are positive constants independent of T .Therefore, F (t) can also be controlled by E(t) from below. Then there exists a

certain positive constant γ such that

d

dtE(t) + γE(t) ≤ 0, for t ∈ [0, T ] (154)

A FULL BIPOLAR HD MODEL FOR SEMICONDUCTORS 1625

Since E(t) is also equivalent to∑2i=1 ‖(ψi, ηi, χi)(t)‖22 by (122), thus solving (154)

by Gronwall inequality yields that

C14

2∑i=1

‖(ψi, ηi, χi)(t)‖22 ≤ E(t) ≤ E(0)e−γt ≤ C15

2∑i=1

‖(ψi, ηi, χi)(0)‖22e−γt,

(155)where C14 and C15 are positive constants independent of T . The inequality (155)and the elliptic estimate (120) yield the desired uniform estimate (85). This com-pletes the proof of Theorem 3.3.

Proof of Theorem 1.2. The existence of the time global solution to the problem (1)-(7) follows from the standard continuation argument with Corollary 1 and Theorem3.3. The decay estimate (12) is derived by (85).

Acknowledgments. The authors would like to thank the anonymous referee forthe important suggestions on improvement of the paper. The first author wouldlike to express his sincere gratitude to Professor Dapeng Du for many importantdiscussions. Zhang’s work is partially supported by the NSFC (No. 11371082) andthe Fundamental Research Funds for the Central Universities (No. 111065201).

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Received December 2013; revised January 2014.

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