AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 213569 10.1155/2014/213569 213569 Research Article Global Existence and Large Time Behavior of Solutions to the Bipolar Nonisentropic Euler-Poisson Equations Chen Min 1 Wang Yiyou 2 Li Yeping 2 Tracinà Rita 1 Department of Mathematics Hubei University of Science and Technology Xianning 437100 China 2 Department of Mathematics Shanghai Normal University Shanghai 200234 China shnu.edu.cn 2014 2712014 2014 06 11 2013 22 12 2013 27 1 2014 2014 Copyright © 2014 Min Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the one-dimensional bipolar nonisentropic Euler-Poisson equations which can model various physical phenomena, such as the propagation of electron and hole in submicron semiconductor devices, the propagation of positive ion and negative ion in plasmas, and the biological transport of ions for channel proteins. We show the existence and large time behavior of global smooth solutions for the initial value problem, when the difference of two particles’ initial mass is nonzero, and the far field of two particles’ initial temperatures is not the ambient device temperature. This result improves that of Y.-P. Li, for the case that the difference of two particles’ initial mass is zero, and the far field of the initial temperature is the ambient device temperature.

1. Introduction

In this paper we study the following 1D bipolar nonisentropic Euler-Poisson equations: (1)n1t+j1x=0,j1t+(j12n1)x+(n1T1)x=n1E-j1,T1t+j1n1T1x+23T1(j1n1)x-23n1T1xx=13(j1n1)2-(T1-T*),n2t+j2x=0,j2t+(j22n2)x+(n2T2)x=-n2E-j2,T2t+j2n2T2x+23T2(j2n2)x-23n2T2xx=13(j2n2)2-(T2-T*),Ex=n1-n2, where ni>0,ji,Ti,(i=1,2), and E denote the particle densities, current densities, temperatures, and the electric field, respectively, and T*>0 stands for the ambient device temperature. The system (1) models various physical phenomena, such as the propagation of electron and hole in submicron semiconductor derives, the propagation of positive ion and negative ion in plasmas, and the biological transport of ions for channel proteins. When the temperature Ti(i=1,2) is the function of the density ni(i=1,2), the system (1) reduces to the isentropic bipolar Euler-Poisson equations. For more details on the bipolar isentropic and nonisentropic Euler-Poisson equations (hydrodynamic models), we can see  and so forth.

Due to their physical importance, mathematical complexity, and wide rang, of applications, many results concerning the existence and uniqueness of (weak, strong, or smooth) solutions for the bipolar Euler-Poisson equations can be found in  and the references cited therein. However, the study of the corresponding nonisentropic bipolar Euler-Poisson equation is very limited in the literature. In  Li investigated the global existence and nonlinear diffusive waves of smooth solutions for the initial value problem of the one-dimensional nonisentropic bipolar hydrodynamic model when the difference of two particles’ initial mass is zero, and the far field of two particles’ initial temperatures is the ambient device temperature. We also mention that there are some results about the relaxation limit and quasineutral limit of the bipolar Euler-Poisson system see . In this paper, we will show the existence and large time behavior of global smooth solutions for the initial value problem of (1), when the difference of two particles’ initial mass is nonzero and the far field of the initial temperatures is not the ambient device temperature. We now prescribe the following initial data: (2)(ni,ji,Ti)(x,t=0)=(ni0,ji0,Ti0)(x),HHHHHHHHHHni0>0,i=1,2,limx±(ni0,ji0,Ti0)(x)=(n±,ji±,Ti±),HHHHHHn±>0,Ti±>0,i=1,2, and (n±,ji±,Ti±) are the state constants. We also give the electric field as x=-; that is, (3)E(-,t)=0. The nonlinear diffusive phenomena both in smooth and weak senses were also observed for the bipolar isentropic and nonisentropic by Gasser et al. , Huang and Li , and Li , respectively. Namely, according to the Darcy’s law, it is expected that the solutions (n1,j1,T1,n2,j2,T2,E)(x,t) converge in L-sense to (n¯,j¯,T*,n¯,j¯,T*,0)(x,t); here (n¯,j¯)=(n¯,j¯)((x+x0)/1+t) (x0 is a shift constants) is the nonlinear diffusion waves, which is self-similar solutions to the following equations: (4)n¯t-(n¯T*)xx=0,j¯=-(n¯T*)x,(n¯,j¯)(n±,0),asx±. Note that in , the author assumed that (5)ji+=ji-,Ti±=T*,i=1,2 which lead to the difference of two particles’ initial mass to be zero; that is, (6)[n10(x)-n20(x)]dx=0,i=1,2. This implies, from the last equation of (1), that (7)E(+,t)-E(-,t)=0. In this paper, we try to drop off these too stiff conditions. That is, ji+ji-,Ti±T*  (i=1,2). Moreover, for stating our results, set for i=1,2, (8)(φi0,ψi0,θi0)(x)=(-x[ni0(ξ)-n^i(ξ,t)-n¯(ξ+x0,t=0)]dξ,-xji0(x)-j^i0(x)-j¯(x+x0,t=0),Ti0(x)-T^i0(x)-T*), where (n^1,j^1,T^1,n^2,j^2,T^2,E^) are the gap functions (or say correction functions) which will be given in Section 2, and (n¯,j¯)=(n¯,j¯)(x+x0,t) are the shifted diffusion waves with x0=(1/(n+-n-))[ni0(x)-n^i0(x)-n¯(x)]  dx for i=1,2.

Throughout this paper, the diffusion waves are always denoted by (n¯,j¯)(x/1+t). C denotes the generic positive constant. Lp()(1p<) denotes the space of measurable functions whose p-powers are integrable on , with the norm ·Lp=(|·|pdx)1/p, and L is the space of bounded measurable functions on , with the norm ·L=esssupx|·|. Without confusion, we also denote the norm of L2() by · for brevity. Hk() (Hk without any ambiguity) denotes the usual Sobolev space with the norm ·k, especially ·0=·.

Now we state our main results as follows.

Theorem 1.

Let (ϕi0,ψi0,θi0)(i=1,2)H3()×H2()×H3(), and set δ=|j1+|+|j1-|+|j2+|+|j2-|+|T1+-T*|+|T1--T*|+|T2+-T*|+|T2--T*|+|n+-n-| and Φ0=(ϕ10,ϕ20)3+(ψ10,ψ20)2+(θ10,θ20)3. Then, there is a δ0>0 such that if Φ0+δδ0 the solutions (n1,n2,j1,j2,θ1,θ2,E) of IVP (1)–(3) uniquely and globally exist and satisfy (9)n1-n^1-n¯,n2-n^2-n¯C([0,+),H2())C1([0,+),H1()),j1-j^1-j¯,j2-j^2-j¯C([0,+),H2())C1([0,+),H1()),T1-T^1-T*,T2-T^2-T*C([0,+),H3())C1([0,+),H1()),E-E^C([0,+),H3())C1([0,+),H2())C2([0,+),H1()). Moreover, it holds that (10)k=02(1+t)k+1xk(k=02n1-n^1-n¯,T1-T^1-T*,n2-n^2-n¯,T2-T^2-T*k=02)(t)2+k=02(1+t)k+2xk(j1-j^1-j¯,j2-j^2-j¯)(t)2+(1+t)3x3(T1-T^1-T*,T2-T^2-T*)(t)2C(δ+Φ0),(n1-n^1-n2+n^2)(t)12+(j1-j^1-j2+j^2)(t)12+(T1-T^1-T2+T^2)(t)22+(E-E^)(t)22C(Φ0+δ)e-αt, for some constant α>0.

Remark 2.

It is more important and interesting that we should discuss the existence and large time behavior of global smooth solution for the bipolar nonisentropic Euler-Poisson system with the general ambient device temperature functions, instead of the constant ambient device temperature, as in . Moreover, we also should consider the similar problem for the corresponding multi-dimensional bipolar non-isentropic Euler-Poisson systems. These are left for the forthcoming future.

The rest of this paper is arranged as follows. In Section 2, we make some necessary preliminaries. That is, we first give some well-known results on the diffusion waves and one key inequality will be used later; then we trickly construct the correction functions to delete the gaps between the solutions and the diffusion waves at the far field. We reformulate the original problem in terms of a perturbed variable and state local-in-time existence of classical solutions in Section 3. Section 4 is used to establish the uniformly a priori estimate and to show the global existence of smooth solutions, while we prove the algebraic convergence rate of smooth solutions in Section 5.

2. Some Preliminaries

In this section, we state the nonlinear diffusive wave and then construct the correction functions. First of all, we list some known results concerning the self-similar solution of the nonlinear parabolic equation (4). Let us recall that the nonlinear parabolic equation (11)n¯t-(n¯T*)xx=0,n¯n±,asx±, possesses a unique self-similar solution n¯(x,t)n¯(ξ),  ξ=x/1+t, which is increasing if n-<n+ and decreasing if n->n+. The corresponding Darcy law is j¯=-(n¯T*)x satisfying j¯0 as x±.

Lemma 3 (see [<xref ref-type="bibr" rid="B5">4</xref>, <xref ref-type="bibr" rid="B14">15</xref>, <xref ref-type="bibr" rid="B4">21</xref>] ).

For the self-similar solution of (11), it holds (12)|n¯(ξ)-n+|ξ>0+|n¯(ξ)-n-|ξ<0,C|n+-n-|e-ν0ξ2,|xktln¯(x,t)|C|n+-n-|(1+t)-(k+2l)/2e-ν0ξ2k+l1,k,l0,-0|n¯(x,t)-n-|2  dx+0+|n¯(x,t)-n+|2  dxC|n+-n-|2(1+t)1/2,|xktln¯|2  dxC|n+-n-|2(1+t)(1/2)-2l-k,k+l1, where ν0>0 is a constant.

Next, we construct the gap function, which will be used in Sections 3 and 4. First of all, motivated by [6, 22], let us look into the behaviors of the solutions to (1)–(3) at the far fields x=±. Then we may understand how big the gaps are between the solution and the diffusion waves at the far fields. Let (ni±(t),ji±(t),Ti±(t))=(ni(±,t),ji(±,t),Ti(±,t)),i=1,2 and E±(t)=E(±,t). From (1)1 and (1)4, since xjix=±=0 for i=1,2, it can be easily seen that (13)ni±(t)=ni(±,t)n±. Differentiating (1)7 with respect to t and applying (1)1 and (1)4, we have Etx=(n1-n2)t=-(j1-j2)x, which implies (14)ddtE+(t)-ddtE-(t)=-[j1+(t)-j2+(t)]+[j1-(t)-j2-(t)]. Taking x=± to (1)2,3 and (1)5,6, we also formally have (15)ddtj1±(t)=n±E±(t)-j1±(t),(16)ddtj2±(t)=-n±E±(t)-j2±(t),(17)ddtTi±(t)=13(ji±ni±)2-(Ti±-T*),  i=1,2. It can be easily seen that (14)–(17) can uniquely determine the unknown state functions ji±(t),Ti±(t)(i=1,2), and E+(t) since we have known E-(t)=E(-,t)=0. Solving these O.D.E and noticing (13), there exists some constant 0<β0<1/2 such that (18)ni(±,t)=n±,i=1,2,|ji(+,t)|=O(1)e-β0t,i=1,2,ji(-,t)=O(1)e-t,i=1,2,Ti(±,t)=T*+(Ti±-T*)e-t+O(1)e-β0t,i=1,2,|E(+,t)|=O(1)e-β0t,E(-,t)=0. Obviously, there are some gaps between ji(±,t) and j¯(±,t), Ti(±,t) and T*, and E(+,t) and E¯0, which lead to ji(x,t)-j¯(x,t),Ti(x,t)-T*,E(x,t)L2(). To delete these gaps, we need to introduce the correction functions (n^1,n^2,j^1,j^2,T^1,T^2,E^)(x,t). As those done in [6, 22], we can construct these gap functions. That is, we can choose (n^1,n^2,j^1,j^2,E^)(x,t), which solve the system (19)n^1t+j^1x=0,j^1t=n˘E^-j^1,n^2t+j^2x=0,j^2t=-n˘E^-j^2,E^x=n^1-n^2, with j^i(x,t)ji±(t) as x±, E^(x,t)0 as x-, and E^(x,t)E+(t) as x+. Here, n˘(x)=n-+(n+-n-)-x+2L0m0(y)dy with m0(x)0,  m0C0(),suppm0[-L0,L0], and -+m0(y)dy=1. Moreover, we take T^i(x,t)=T^i-(t)(1-g(x))+T^i+(t)g(x)(i=1,2) with g(x)=-xm0(y)  dy, which together with (17) implies (20)tT^i(x,t)=-T^i(x,t)+13(ji-(t)n-)2(1-g(x))tT^i(x,t)=+13(ji+(t)n+)2g(x)tT^i(x,t)=:-T^i(x,t)+Si(x,t),i=1,2.

In conclusion, we have constructed the required correction functions (n^1,n^2,j^1,j^2,T^1,T^2,E^) which satisfy (21)n^1t+j^1x=0,j^1t=n˘E^-j^1,n^2t+j^2x=0,j^2t=-n˘E^-j^2,T^it=-T^i+Si(x,t),E^x=n^1-n^2,with{j^i(x,t)ji±(t),as  x±,T^i(x,t)Ti±(t)-T*,as  x±,E^(x,t)0,as  x-,E^(x,t)E+(t),as  x+. Since these details can be found in [6, 22], we only give the following decay time-exponentially of (n^1,n^2,j^1,j^2,T^1,T^2,E^)(x,t).

Lemma 4.

There exist positive constants C and ν<1/2 independent of t, such that (22)(n^i,j^i,T^i,E^)(t)L()Cδe-νt,      i=1,2,and suppn^i=suppm0[-L0,L0],  i=1,2.

3. Reformulation of Original Problem

In this section, we first reformulate the original problem in terms of the perturbed variables. Setting for i=1,2, (23)(φi,ψi,θi,)(x,t)=(-x[ni(ξ,t)-n^i(ξ,t)-n¯(ξ+x0,t)]  dξ,ji(x,t)-j^i(x,t)-j¯(x+x0,t),Ti(x,t)-x-T^i(x,t)-T*,E(x,t)-E^(x,t)),then from (1), (11), and (21), we have for i=1,2, (24)φit+ψi=0,ψit+((-φit+j^i+j¯)2φix+n^i+n¯+(φix+n^i+n¯)(-φit+j^i+j¯)2φix+n^i+n¯×(θi+T^i+T*)-n¯T*)x=(-1)i-1(φix+n^i+n¯)+(-1)i-1(φix+n^i+n¯-n˘)E^-ψi+(n¯T*)tx,θit+-φit+j^i+j¯φix+n^i+n¯(θi+T^i+T*)x+23(-φit+j^i+j¯φix+n^i+n¯)x(θi+T^i+T*)-23(φix+n^i+n¯)(θi+T^i+T*)xx=[13(-φit+j^i+j¯φix+n^i+n¯)2-Si(x,t)]-θi,=φ1-φ2,

with the initial data (φi,ψi,θi)(x,0)=(φ10,ψi0,θi0)(x), i=1,2. Further, we have(25)φ1tt+φ1t-((θ1+T^1+T*)φ1x+n¯θ1)x+(φ1x+n^1+n¯)=-f1+g1x-(n¯T*)tx,φ2tt+φ2t-((θ2+T^2+T*)φ2x+n¯θ1)x-(φ2x+n^2+n¯)=f2+g2x-(n¯T*)tx,θ1t+θ1-23(φ1x+n^1+n¯)θ1xx-23(θ1+T^1+T*)(φ1tφ1x+n^1+n¯)x=G1,θ2t+θ2-23(φ2x+n^2+n¯)θ2xx-23(θ2+T^2+T*)(φ2tφ2x+n^2+n¯)x=G2,

with the initial data (26)φi(x,0)=φi0(x),φit(x,0)=-ψi0(x),θi(x,0)=θi0(x),i=1,2. Here (27)fi=(φix+n^i+n¯-n˘)E^-((θi+T^i+T*)n^i+n¯T^i)x,gi=(-φit+j^i+j¯)2φix+n^i+n¯,Gi=--φit+j^i+j¯φix+n^i+n¯(θi+T^i+T*)xGi=-23(j^i+j¯φix+n^i+n¯)x(θi+T^i+T*)-23(j^i+j¯φix+n^i+n¯)x(θi+T^i+T*)-Si(x,t).

By the standard iteration methods (see ), we can prove the local existence of classical solutions of the IVP (25) and (26). Here for the sake of clarity, we only state result and omit the proof.

Lemma 5.

Suppose that (φi0,-ψi0,θi0)H3()×H2()×H3() for i=1,2. Then there is a C1>0 such that if (28)(φ10,θ10,φ20,θ20)32+(ψ10,ψ20)22C1,

then there is a positive number T0 such that the initial value problems (25) and (26) have a unique solution (φ1,θ1,φ2,θ2) satisfying φiC([0,T0];H3())C1([0,T0];H2())C2([0,T0]; H1()),θiC([0,T0];H3())C1([0,T0];H1())(i=1,2), and (29)(φ1,θ1,φ2,θ2)(·,t)32+(φ1t,φ2t)(·,t)22+(θ1t,θ2t)(·,t)12C,

for some positive constant C.

To end this section, we also derive (30)tt+t+2n¯-(n¯(θ1-θ2)+(θ1+T^1+T*)x)x=h1x-h2-h3+h4x,(31)(θ1-θ2)t+(θ1-θ2)-23(φ1x+n^1+n¯)(θ1-θ2)xx-2(θ1+T^1+T*)3(φ1x+n^1+n¯)tx=G3,

where (32)h1:=(θ1+T^1+T*)(n^1-n^2)+(φ2x+n^2)(θ1-θ2)+(φ2x+n^2+n¯)(T^1-T^2),h2:=(φ1x+φ2x+n^1+n^2),h3=[φ1x+φ2x+n^1+n^2+2(n¯-n˘)]E^,h4=(-φ1t+j^1+j¯)2φ1x+n^1+n¯-(-φ2t+j^2+j¯)2φ2x+n^2+n¯,G3:=G1-G2+[23(φ1x+n^1+n¯)-23(φ2x+n^2+n¯)]θ2xx+[2(θ1+T^1+T*)3(φ1x+n^1+n¯)-2(θ2+T^2+T*)3(φ2x+n^2+n¯)]φ2tx-2(θ1+T^1+T*)3(φ1x+n^1+n¯)2φ1txx-2(θ1+T^1+T*)3(φ1x+n^1+n¯)2n^1xt+[2(θ2+T^2+T*)3(φ2x+n^2+n¯)2φ2t-2(θ1+T^1+T*)3(φ1x+n^1+n¯)2φ1t]×(φ2x+n¯)x-[2(θ2+T^2+T*)3(φ2x+n^2+n¯)2n^1x-2(θ1+T^1+T*)3(φ1x+n^1+n¯)2n^2x]φ2t.

4. Global Existence of Smooth Solutions

In this section we mainly prove global existence of smooth solutions for the initial value problems (25) and (26). To begin with, we focus on the a priori estimates of (φ1,θ1,φ2,θ2). For this purpose, letting T(0,+), we define (33)X(T)={(φi,φit,θi,θit):tjφiC([0,T];H3-j()),θiC([0,T];H3()),θitC([0,T];H1()),([0,T];H3-j())i=1,2,j=0,1}, with the norm (34)N(T)2=max0tT{(φ1,φ2,θ1,θ2)(t)32+(φ1t,φ2t)(t)22N(T)2=max0tT+(θ1t,θ2t)(t)12}. Let N(T)2ɛ2, where ɛ is sufficiently small and will be determined later. Then, by Sobolev inequality, we have for i=1,2, (35)(φi,φix,φixx,θi,θix,θit,θixx,φit,φitx)(t)LCɛ. Clearly, there exists a positive constant c1,c2 such that (36)0<1c1ni=φix+n^i+n¯c1,0<1c2Ti=θi+T^i+T*c2,i=1,2. Further, from (24)7, we also have tjC(0,T;H2-j()) and (37)(,x,t)(t)LCɛ.

Now we first establish the following basic energy estimate.

Lemma 6.

Let (φ1,θ1,φ2,θ2)(x,t)X(T) be the solution of the initial value problem (25) and (26). If δ+ɛ1, then it holds that for 0<t<T, (38)i=02(φi,φix,φit,θi)(·,t)2+(·,t)2+0t(·,τ)2  dτ+i=120t(φix,φit,θi,θix)(·,τ)2  dτC(Φ0+δ).

Proof.

Multiplying (25)1 and (25)2 by φ1 and φ2, respectively, and integrating them over by parts, we have for i=1,2, (39)ddt(φiφit+12φi2)dx+(θi+T^i+T*)φix2  dx+(-1)i-1(φix+n^i+n¯)  φidx-φit2  dx=-n¯θiφix  dx+(n¯T*)tφix  dx+(-1)ifiφidx-giφixdx. Using Cauchy-Schwartz’s inequality, and Lemmas 3 and 4, we have (40)-n¯θiφix  dx+(n¯T*)tφix  dxκφix2  dx+C(θi2+n¯t2)dx, where and in the subsequent κ>0 is some proper small constant, and (41)(-1)ifiφi  dxCɛφix2  dx+Cδ2(1+t)1/4e-νt, where we also used the facts (42)(n¯-n˘)2  dxCδ2(1+t)1/2 which can be proved from the construction of n˘(x)n±, as x±, and the property of the diffusion wave n¯((x+x0)/(1+t)). Similarly, we can show (43)-giφixdx=-(-φit+j^i+j¯)2φix+n^i+n¯φixdxC(δ+ɛ)(φix2+φit2)dx+Cδn¯x2  dx+Cδ2e-νt,

which together with (39)–(41) implies, (44)ddt(φiφit+12φi2)dx+[(θi+T^i+T*)-κ]φix2  dx-φit2  dx+(-1)i-1(φix+n^i+n¯)φi  dxC(δ+ɛ)(φix2+φit2)dx+C(θi2+n¯t2+n¯x2)dx+Cδ2e-ν1t,

where 0<ν1<ν. Moreover, for the coupled term with the electric field, we have (45)((φ1x+n^1+n¯)φ1-(φ2x+n^2+n¯)φ2)dxn¯2  dx-Cɛ(2+φ1x2+φ2x2)dx-Cδ2e-νt.

Next, multiplying (25)1 and (25)2 by φ1t and φ2t, respectively, and integrating their sum over by parts, we have (46)ddt(12φit2+12(θi+T^i+T*)φix2)dx+(φit2+n¯θiφitx+(-1)i-1(φix+n^i+n¯)φit)dx=(-1)ifiφitdx+gixφitdx-[(n¯T*)txφit-12(θi+T^i+T*)tφix2]dx. Using Schwartz’s inequality, (42), and Lemmas 3 and 4, we have (47)-[(n¯T*)txφit-12(θi+T^i+T*)tφix2]dx+(-1)i-1fiφit  dxκφit2  dx+C(δ+ɛ)(φit2+φix2)dx+Cn¯tx2  dx+Cδ2(1+t)1/4e-νt.

Since (48)gix=-(-φit+j^i+j¯)2(φix+n^i+n¯)2φixx-2(-φit+j^i+j¯)(φix+n^i+n¯)φixt+O(1)[(φit+j^i+j¯)(n^i+n¯)t+(j^i+j¯)2(n^i+n¯)x+(j^i+j¯)2(n^i+n¯)xφit2],

we obtain, after integration by parts, that (49)gixφit  dxddt(-φit+j^i+j¯)22(φix+n^i+n¯)2φix2  dx+C(δ+ɛ)(φit2+φix2)dx+Cδ(n¯t2+n¯x4)dx+Cδ2e-νt, where we have used (50)((-φit+j^i+j¯)2(φix+n^i+n¯)2)xL,((-φit+j^i+j¯)2(φix+n^i+n¯)2)tLC(δ+ɛ),

with the aid of |φitt|<C|φixx+φixt+φix+φit+φi+θi+θix+n¯xt|+Cδ2e-νt. Putting the above inequality into (46), we have (51)ddt[12φit2+12(θi+T^i+T*)φix2-(-φit+j^i+j¯)22(φix+n^i+n¯)2φix2]dx+(1-κ)φit2  dx+(-1)i-1(φix+n^i+n¯)φit  dx+n¯θiφitx  dxC(δ+ɛ)(φit2+φix2)dx+C(n¯tx2+n¯t2+n¯x4)dx+Cδ2e-ν1t. On the other hand, we have (52)((φ1x+n^1+n¯)φ1t-(φ2x+n^2+n¯)φ2t)dxddt12n¯2  dx-12n¯t2dx-Cɛ(φ1x2+φ1t2+φ2x2+φ2t2)dx-Cδe-νt.

Finally, multiplying (25)l  (l=3,4) by (3n¯(φix+n^i+n¯)/2(θi+T^i+T*))θi  (i=1,2) and integrating the resultant equation by parts over , we have (53)ddt3n¯(φix+n^i+n¯)4(θi+T^i+T*)θi2dx+[3n¯(φix+n^i+n¯)2(θi+T^i+T*)θi2+n¯θi+T^i+T*θix2]dx,-n¯θiφitxdx=(3n¯(φix+n^i+n¯)4(θi+T^i+T*))tθi2  dx-n¯θiφitxdx=-(n¯θi+T^i+T*)xθiθix  dx-n¯θiφitxdx=-n¯θiφix+n^i+n¯φit(φix+n^i+n¯)x  dx-n¯θiφitxdx=+Gi3n¯(φix+n^i+n¯)2(θi+T^i+T*)θi  dx. Now we estimate the term of the right hand side of (53), using Cauchy-Schwartz’s inequality and Lemmas 3 and 4. First, with the help of the following equality j^i=(1-g(x))ji-(t)+g(x)ji+(t) (see [6, 22]), we have (54)3n¯(φix+n^i+n¯)2(θi+T^i+T*)(j^i23(φix+n^i+n¯)2-Si(x,t))θi  dx={--L0+L0++-L0L0}3n¯(φix+n^i+n¯)2(θi+T^i+T*)×(j^i23(φix+n^i+n¯)2-Si(x,t))θi  dx=--L03n¯(φix+n^i+n¯)2(θi+T^i+T*)θij^i2(13ni2-13n-2)dx+L0+3n¯(φix+n^i+n¯)2(θi+T^i+T*)×θij^i2(13ni2-13n+2)  dx+-L0L03n¯(φix+n^i+n¯)2(θi+T^i+T*)θi×[j^i23ni2-(1-g(x))3n-2(j-(t))2-g(x)3n+2(j+(t))2]dxCδ(φix2+θi2)  dx+Cδ2e-νt--L0(n¯-n-)2  dx+Cδ2e-νtL0+(n¯-n+)2  dx+Cδ2e-νtCδ(φix2+θi2)  dx+Cδ2(1+t)1/2e-νt, which implies (55)3n¯(φix+n^i+n¯)2(θi+T^i+T*)((-φit+j^i+j¯)23(φix+n^i+n¯)2-Si(x,t))θi  dx=3n¯(φix+n^i+n¯)2(θi+T^i+T*)(φit23ni2-2φit(j^i+j¯)3ni2)θi  dx+3n¯(φix+n^i+n¯)2(θi+T^i+T*)j¯2+2j^ij¯3ni2θi  dx+3n¯(φix+n^i+n¯)2(θi+T^i+T*)θi(j^i23ni2-Si(x,t))dxC(δ+ɛ)(φix2+φit2+θi2)dx+Cδn¯x2  dx+Cδ2(1+t)1/2e-νt. From the definition of Gi  (i=1,2), and using Schwartz’s inequality, we have (56)Gi3n¯(φix+n^i+n¯)2(θi+T^i+T*)θi  dxκθi2  dx+C(δ+ɛ)(θi2+θix2+φix2+φit2)dx+C(n¯x2+n¯xx2)dx+Cδ2e-ν2t, with ν1<ν2<ν. And using Schwartz’s inequality and Lemma 3 yields (57)(3n¯(φix+n^i+n¯)4(θi+T^i+T*))tθi2  dx-(n¯θi+T^i+T*)xθiθix  dxC(δ+ɛ)(θi2+θix2)dx,-n¯θiφix+n^i+n¯φit(φix+n^i+n¯)xdxddtn¯θi2(φix+n^i+n¯)φix2dx+C(δ+ɛ)(θi2+φix2+φit2)dx. Putting the above inequalities into (53) yields (58)ddt[3n¯(φix+n^i+n¯)4(θ1+T^i+T*)θi2-n¯θi2(φix+n^i+n¯)φix2]dx+[(3n¯(φix+n^i+n¯)2(θi+T^i+T*)-κ)θi2+n¯θi+T^i+T*θix2]dx-n¯θiφitx  dxC(δ+ɛ)(θi2+θix2+φit2+φix2)dx+C(n¯x2+n¯xx2)dx+Cδ2e-ν2t. Combining (44), (45), (51), (52), and (58), we can obtain (38); this completes the proof.

Further, in the completely similar way, we can show the following.

Lemma 7.

Let (φ1,θ1,φ2,θ2)(x,t)X(T) be the solution of the initial value problems (25) and (26); then it holds that for 0<t<T, (59)i=12(φix,φixx,φitx,φitt,θix,θit,θixx)(·,t)2+x(·,t)2+0t(i=12(φixx,φitx,φitt,θix,θit,θixx,θitx,θixxx)(·,τ)2i=12+x(·,τ)2)  dτC(Φ0+δ),i=12(φixx,φixxx,φitxx,φittx,θixx,θitx,θixxx)(·,t)2+xx(·,t)2+0t(i=12(φixx,φix,φitx,θixx,θitx,θixxx)(·,τ)2i=12+xx(·,τ)2)  dτC(Φ0+δ),

provided that ɛ+δ1.

Based on the local existence given in Lemma 5 and the a priori estimates given in Lemmas 6 and 7, by the standard continuity argument, we can prove the global existence of the unique solutions of the IVP (25) and (26).

Theorem 8.

Under the assumption of Theorem 1, the classical solution (φ1,θ1,φ2,θ2,)(x,t) of the solutions of the IVP (25) and (26) exist globally in time if Φ0+δ is small enough. Moreover, one has (60)(φ1,θ1,φ2,θ2)(·,t)32+(φ1t,φ2t,(·,t))22+(θ1t,θ2t)(·,t)12+0t((φ1x,φ1t,φ2x,φ2t,)(·,τ)22+(θ1,θ2)(·,τ)42+(θ1t,θ2t)(·,τ)22)  dτC((φ10,θ10,φ20,θ20)32+(ψ10,ψ20)22+δ),t>0,(φ1,θ1,φ2,θ2)(·,t)32+(φ1t,φ2t,(·,t))22+(θ1t,θ2t)(·,t)120,t.

5. The Algebraic Decay Rates

In this section, we prove the time-decay rate of smooth solutions (φ1,θ1,φ2,θ2) of (25) with the initial data (φ10,-ψ10,θ10,φ20,-ψ20,θ20). For this aim, using the idea of [4, 15, 24], we first prove the exponential decay of and θ1-θ2 to zero then obtain the algebraic convergence of (φ1,θ1,φ2,θ2). Due to Theorem 8, we know that the global smooth solutions (φ1,θ1,φ2,θ2) satisfy (61)(φ1,θ1,φ2,θ2)32+(φ1t,φ2t,)22+(θ1t,θ2t,t)12C(Φ0+δ), which leads to, in terms of Sobolev embedding theorem, that (62)(φ1,φ2,φ1x,φ2x,φ1xx,φ2xx,φ1t,φ1tx,φ2t,φ2tx,θ1,θ1x,θ1xx,θ2,θ2x,θ2xx,,x,t)L()C(Φ0+δ). Further, by (25), we also have (63)(φ1tt,φ2tt,θ1t,θ2t)L()C(Φ0+δ).

Lemma 9.

Let (φ1,θ1,φ2,θ2) be the global classical solutions of IVP (25) and (26) satisfying Φ0+δ1. Then it holds for and θ1-θ2 that for t>0, (64)(,x,t,xx,tx,θ1-θ2,(θ1-θ2)x,(θ1-θ2)xx)(·,t)C(Φ0+δ)e-γ0t.

Proof.

Multiplying (30) by and integrating it by parts over , we obtain (65)ddt(t+122)dx-t2  dx+2n¯2  dx+(θ1+T^1+T*)x2  dx=-n¯(θ1-θ2)x  dx+(h1x-h2-h3+h4x)  dx. Using Cauchy-Schwartz’s inequality, Lemmas 3 and 4, (62), and (63), we have (66)-n¯(θ1-θ2)x  dx+h1x  dxκx2  dx+C(θ1-θ2)2  dx+Cδe-νt,-(h2+h3)  dxC(Φ0+δ)2  dx+Cδe-ν1t. Moreover, noticing that (67)h4x=-j12n12xx-2j1n1tx+O(1)(n^1x+n^2x+j^1x+j^2x)h4x=+O(1)(φ2xx+φ2tx+n¯x+j¯x+n^2x+j^2x)h4x=×(x+t+n^1+n^2+j^1+j^2),

then (68)h4x  dxC(Φ0+δ)(2+x2+t2)dx+Cδe-νt. Therefore, we have (69)ddt(t+122)dx-t2  dx+2n¯2  dx+[(θ1+T^1+T*)-κ]x2  dxC(Φ0+δ)(2+x2+t2)dx+C(θ1-θ2)2  dx+Cδe-ν1t. While multiplying (30) by t and integrating the resultant equation by parts over , similarly, we can show (70)ddt(12t2+n¯2+(12(θ1+T^1+T*)-j12n12)x2)dx+t2  dx+n¯(θ1-θ2)tx  dxC(Φ0+δ)×((θ1-θ2)2+(θ1-θ2)x2+2+x2+t2)dx+Cδe-ν1t.

Next, multiplying (31) by (3n¯(φ1x+n^1+n¯)/2(θ1+T^1+T*))(θ1-θ2) and integrating the resultant equation by parts over , we get (71)ddt3n¯(φ1x+n^1+n¯)4(θ1+T^1+T*)(θ1-θ2)2  dx+(3n¯(φ1x+n^1+n¯)2(θ1+T^1+T*)(θ1-θ2)23n¯(φ1x+n^1+n¯)2(θ1+T^1+T*)+n¯θ1+T^1+T*(θ1-θ2)x2)dx-n¯(θ1-θ2)tx  dx=(3n¯(φ1x+n^1+n¯)4(θ1+T^1+T*))t(θ1-θ2)2  dx-(n¯θ1+T^1+T*)x(θ1-θ2)(θ1-θ2)x  dx+G33n¯(φ1x+n^1+n¯)2(θ1+T^1+T*)(θ1-θ2)dxC(Φ0+δ)×((θ1-θ2)2+(θ1-θ2)x2+x2+t2)dx+C(Φ0+δ)e-ν2t,

where in the last inequality, we have used (72)(3n¯(φ1x+n^1+n¯)4(θ1+T^1+T*))t(θ1-θ2)2dx-(n¯θ1+T^1+T*)x(θ1-θ2)(θ1-θ2)xdxC(Φ0+δ)((θ1-θ2)2+(θ1-θ2)x2)dx,(73)G33n¯(φ1x+n^1+n¯)2(θ1+T^1+T*)(θ1-θ2)dxC(Φ0+δ)((θ1-θ2)2+(θ1-θ2)x2+x2+t2)+C(Φ0+δ)e-ν2t,

with the aid of (74)i=12(-1)i-1((-φit+j^i+j¯φix+n^i+n¯)2-Si(x,t))×3n¯(φ1x+n^1+n¯)2(θ1+T^1+T*)(θ1-θ2)dxC(Φ0+δ)((θ1-θ2)2+x2+t2)dx+C(Φ0+δ)e-ν2t.

Combine (69), (70), and (71), and choose proper positive constant λ1 and Λ1 such that (75)λ1×(70)+Λ1×((71)+(72))~t2+2+x2+(θ1-θ2)2.

Then, we have (76)ddt(t,,x,(θ1-θ2))(·,τ)2+C(t,,x,(θ1-θ2),(θ1-θ2)x)(·,t)2C(Φ0+δ)e-ν2t,

which, together with Gronwall’s inequality, yields (77)(,x,t,(θ1-θ2))(·,t)2C(Φ0+δ)e-γ1t, for some positive constants γ1 and C. In the completely same way, treating λ2(30)xx+Λ2((30)xtx+(31)x(3n¯(φ1x+n^1+n¯)/2(θ1+T^1+T*))(θ1-θ2)x)dx for proper positive constants λ2 and Λ2, we can show (78)(x,xx,tx,(θ1-θ2)x(·,t)2C(Φ0+δ)e-γ2t,

for some constant γ2.

Moreover, from (30), (77), and (78), we obtain (79)tt2C(Φ0+δ)e-γ3t,

for γ3=min{γ1,γ2}. Finally, by (31)t(θ1-θ2)t  dx and using (77)–(79), there is a positive constant γ4 such that (80)(θ1-θ2)t2C(Φ0+δ)e-γ4t,

while from (31) and (77)–(80), we have (81)(θ1-θ2)xx2C(Φ0+δ)e-γ5t,

with γ5=min{γ3,γ4}. Combination of (77)–(80) and (81) yields (64). This completes the proof.

In the following, using the idea of [4, 15], we turn to derive the time-decay rate of (φ1,θ1,φ2,θ2) by which we are able to obtain the algebraical decay rate of (φ1,θ1,φ2,θ2) in large time.

Lemma 10.

Let (φ1,θ1,φ2,θ2) be the global classical solution of the IVP (25) and (26) with initial data satisfying Φ0+δ1. If it holds for (φ1,φ2,θ1,θ2)(t>0) that (82)k=03(1+t)kxk(φ1,φ2)2+k=02(1+t)k+2xk(φ1t,φ2t)2+k=01(1+t)k+2xk(θ1t,θ2t)2+k=02(1+t)k+1xk(θ1,θ2)2+(1+t)3x3(θ1,θ2*)21,

then one has (83)k=03(1+t)kxk(φ1,φ2)2+k=130t(1+τ)kxk(φ1,φ2)2  dτ+k=02(1+t)k+1xk(θ1,θ2)2+k=020t(0t(1+τ)k+1xk(θ1,θ2)2xk(θ1,θ2)2+(1+τ)3x3(θ1t,θ2t)20t)  dτC(Φ0+δ),k=02(1+t)k+2xk(φ1t,φ2t)2+k=020t(1+τ)k+2xk(φ1τ,φ2τ)2  dτ+k=01(1+t)k+2xk(θ1t,θ2t)2+(1+t)3x3(θ1t,θ2t)2+0t(k=01(1+τ)k+2xk(θ1,θ2)2k=01+(1+τ)3x3(θ1τ,θ2τ)2)dτC(Φ0+δ).

Since the proof is similar as that in , we can omit the details.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research is partially supported by the National Science Foundation of China (Grant no. 11171223), the Ph.D. Program Foundation of Ministry of Education of China (Grant no. 20133127110007), and the Innovation Program of Shanghai Municipal Education Commission (Grant no. 13ZZ109).

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