# The Global Existence and Uniqueness of the Classical Solution with the Periodic Initial Value Problem for One-Dimension Klein-Gordon-Zakharov Equations

Authors: Cong Sun; Lixia Li
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

Through applying Galerkin method, we establish the approximating solution for one-dimension Klein-Gordon-Zakharov equations and obtain the local classical solution. By applying integral estimates, we also obtain the existence and uniqueness of the global classical solution of Klein-Gordon-Zakharov equations.


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## Body

## 1. Introduction

The Klein-Gordon-Zakharov equations are used to show the interaction between Langmuir wave and ion wave in the plasma, where(1)utt-uxx+u+nu=0,ntt-nxx=u2xx,with periodic initial condition(2)ux,0=u0,nx,0=n0.Here, u(x,t) (complex function) denotes the biggest moment scale component produced by electron in electric field. n(x,t) (real function) denotes the speed of deviations between the ion at any position and that at equilibrium position. x∈R; 0≤t≤T.Inspired by B. L. Guo’s idea and method [1], our paper focuses on the existence and uniqueness of the local and global classical solution of the periodic initial value problem for the above one-dimension Klein-Gordon-Zakharov equations. In past decades, many scholars devoted their researches to Zakharov system and have obtained some important results [2–4].In [5], H. Pecher has researched the following Klein-Gordon equation:(3)ϕtt-Δϕ+ϕ=ϕϕp-1,where ϕ=ϕ(x,t), x∈Rn, t≥0. Given the existence of the Cauchy problem of (3), H. A. Levine in [6] adopted potential well method and concave function method to study the blowing up of the solution of the Cauchy problem of this equation. Moreover, in [7], W. A. Strauss introduced a compactness lemma, which proved existence of the solitary wave of these equations. In [4], Ozawa, Tsutaya, and Tsutsumi studied the following Klein-Gordon-Zakharov system:(4)ϕtt-Δϕ+ϕ=-ψϕ,ψtt-c2Δψ=Δϕ2,with initial condition(5)ϕx,0=ϕ0,ϕtx,0=ϕ1,ψx,0=ψ0,ψtx,0=ψ1.Through applying harmonic analysis method and contracting mapping principle, they obtained that the Cauchy problem (4)-(5) had local well-posedness about t in energy space H1(R3)×L2(R3). Furthermore, based on conservation of energy, they also obtained global existence and uniqueness with small initial value of above Cauchy problem.Furthermore, in [8], T. C. Wang et al considered the following Klein-Gordon-Zakharov equations:(6)utt-uxx+u+nu+u2u=0,ntt-nxx=u2xx.By applying Leray-Schauder fixed-point theorem, they proved the existence of different solutions of system (6).Recently, Z. Y. Zhang et al. studied [9] the following Klein-Gordon-Zakharov equations:(7)utt-uxx+u+αnu=0,ntt-nxx=βu2xx.With different parameter conditions, they adopted the bifurcation method and dynamical systems approach to research bifurcation analysis and dynamic behaviour of travelling wave solutions of (7).Besides, in [10], by combining the trigonometric function series method and the exp-function method, Y. Zhang et al. considered the travelling wave solutions of (7).Now, let us describe our results. Inspired by [1], we first introduced the following equations:(8)nt=ϕxx,(9)ϕt=n+u2,(10)ut=ψxx,(11)ψxxt=uxx-u-nu,with periodic initial data(12)nx,0=n0,ϕx,0=ϕ0,ux,0=u0,ψx,0=ψ0.Moreover, n(x,t), ϕ(x,t), u(x,t), ψ(x,t), n(x,0), ϕ(x,0), u(x,0), and ψ(x,0) are periodic functions with L about x(-∞<x<+∞,L>0,0≤t≤T). Firstly, applying Galerkin method, we structure the approximate solution of (8)–(11); the aim is to obtain the local classical solution. Next, we employ this method of integral estimates to prove the existence and uniqueness of the global solution of (1)-(2).We divide this paper into four parts. In Section2, we establish a priori estimate of (8)–(11). In Section 3, we give the existence of local and global smooth solutions of the above equations. Lastly, in Section 4, we prove existence and uniqueness of the global classical solution of (1)-(2).Notation. In complex Hilbert function space, inner product and standard norm are defined as follows: (13)f,g=∫0Lfxgx¯dx,fL2=f=∫0Lf2dx,where g(x)¯ is conjugate function of g(x).H s is complex function space with generalized differentiable function Dku(k≤s)∈L2, and the norm is defined as follows: (14)fs2=∑k≤sDkfL22.L s ( 0 , T ; H s ) denotes function space about complex function of f(x,t). Moreover, for any 0≤t≤T, f(x,t)∈Hs, and sup0≤t≤TfHs is bound. Particularly, (15)fx,tL∞=sup⁡fx,t.Furthermore,C0,C1,C2,…,D0,D1,D2,…,E0,E1,E2, … are estimate constants, which relate to initial conditions.


## 2. A Priori Estimate of Integrating

Lemma 1.
Ifu0∈L2, ψxx≤C0u, and E0=u02, then u2≤C1E0.Proof.
Equation (10) makes inner product with u and takes real part; we have (16)∫0L12uutdx=∫0Lψxxu¯dx≤∫0Lψxxu¯dx,12ddtu2≤C0u2.By Gronwall inequality, we can obtain that(17)u2≤C1E0.Lemma 2.
Ifu0t,u0x,u0,n0x,ϕ0x,ψ0x∈L2, then(18)ut2+ux2+u2+∫0Lnu2dx+12∫0Lϕx2dx+12∫0Ln2dx=M1,where (19)u0t2+u0x2+u02+∫0Ln0u02dx+12∫0Lϕ0x2dx+12∫0Ln02dx=M1.Proof.
Taking inner product (11) with ut and making real part, we have that(20)∫0Lψxxtut¯dx=∫0Luttut¯dx=12ddt∫0Lut2dx=12ddtut2,(21)∫0Luxxut¯dx=-12ddt∫0Lux2dx=-12ddtux2,(22)∫0Luut¯dx=12ddt∫0Lu2dx=12ddtu2,(23)∫0Lnuut¯dx=12ddtnu2dx+14ddt∫0Lϕx2dx+14ddt∫0Ln2dx.
Combining (20)–(23) yields (24)ddtut2+ux2+u2+∫0Lnu2dx+12∫0Lϕx2dx+12∫0Ln2dx=0.
Therefore,(25)u0t2+u0x2+u02+∫0Ln0u02dx+12∫0Lϕ0x2dx+12∫0Ln02dx=M1.Lemma 3.
Ifu0∈H1, ϕ0∈H1, and n0∈L2, then (26)uH12≤E1,n2≤E2,ϕH1≤E3,ϕt2≤E4,where E1, E2, E3, and E4 are constants, which only relate to n0, u0H1, ϕ0H1, and T.Proof.
From (18), we can easily obtain that(27)ux2+u2≤M1.So that,(28)uH1≤E1.By (9) and (18), we have(29)∫0Ln2dx=∫0Lϕt-u2dx≤2∫0Lϕt2+u4dx≤2M1.Hence, (30)n2≤2M1=E2,ϕt2≤E4.Furthermore, (9) takes inner product with ϕ; we get (31)ddtϕ2≤ϕ2+2n2+u22≤ϕ2+E2+M1≤E5ϕ2.By Gronwall inequality, we have (32)ϕ2≤ϕ02eT.Hence, in combination with (18), we can deduce that (33)ϕH1≤E3.Lemma 4.
Ifu0∈H1, u0t∈H2, u0x∈H2, and n0∈H1, then (34)ut2+ux2+u2≤E5.Proof.
The first equation of (1) makes inner product with ut and takes real part; we have that(35)ddtut2+ux12+u2≤12∫0Tn4dx+∫0Tu4dx+2ut2≤12n2nL∞2+12u2uL∞2+ut2≤α12nx2+c12n2n2+α22u2+c22ux2u2+ut2≤B1ut2+ux2+u2+B2,where B1=max⁡{1,α22C1E0,c22C1E0} and B2=α12E1nx2.
So that, by Gronwall inequality, we obtain that(36)ut2+ux2+u2≤E5.Corollary 5.
u L ∞ 2 ≤ E 51, utL∞2≤E52, uxL∞2≤E53, and nL∞≤E54.Lemma 6.
Ifn0tt,n0xt,u0tt,u0xx∈L2, then one has (37)ntt2+nxt2+utt2+uxx2≤E6.Proof.
We differentiatet with respect to the second equation of (1); we obtain that(38)nttt-nxxt=u2xxt.Equation (38) takes inner product with ntt; we have that(39)ddtntt2+ddtnxt2≤ntt2+u2xxt2.Furthermore, (40)uxxt2=4utxux¯+2utxxu¯+2uxxut¯≤4ux¯utxL∞+2utxxu¯L∞+2uxxutL∞≤4c11u1-α12utxL∞+2E51utxx+2E52uxx≤4c11C1E01-α12c12E5+α13utxx+2E51utxx+2E52uxx≤4c11C1E01-α12c12E5+4c11C1E0α131-α12c131-α14uxx+2E51c13uxx1-α14+2uxxE52≤B31+uxx,where B3=max⁡{4c11C1E0c12E5/1-α12,4c11c13C1E0α13/(1-α12)(1-α14)+2E51c13/1-α14+2E52}.
So,(41)uxxt22≤B321+uxx2.
Differentiating (1) with respect to t, we have that(42)uttt-uxxt+ut+nut+ntu=0.
Next, (42), respectively, makes inner product with uxx and utt and takes real part; we can obtain that(43)ddtuxx2≤uxx2+uttt2+2utL∞2uxx2+utL∞n2+ux2+ntx2+u2ux2≤uxx2+c1421-α142utt2+2E5uxx2+E5ntx2+E53/2+E5M1+E5C1E0≤B4uxx2+utt2+ntx2+1,(44)ddtutt2≤uxxt¯2+2utt2+nL∞ut2+utt2+ntL∞u2+utt2≤c1521-α152uxx2+3+E2utt2+α16ntt2+C1E0+E2E5+c162E2≤B5uxx2+utt2+ntt2+1.Here, (45)B4=max⁡1+2E5,c1421-α142,E5,E53/2+E5M1+E5C1E0,B5=max⁡c1521-α152,3+E2,C1E0+E2E5+c162E2.
Combining (39), (41), (43), and (44) yields (46)ddtntt2+nxt2+utt2+uxx2≤B6ntt2+nxt2+utt2+uxx2+1.Therefore, applying Gronwall inequality, we can obtain (47)ntt2+nxt2+utt2+uxx2≤E6.Lemma 7.
Ifn0ttt2+n0xtt2+u0ttt2≤E7, then (48)nttt2+nxtt2+uttt2≤E8.Proof.
Differentiating (8) with respect to t thrice and differentiating (9) with respect to x and t twice and eliminating ϕxxtt, we have(49)ntttt-nxxtt=uxxtt2.Furthermore, also differentiating (10) with respect to t thrice and differentiating (11) with respect to t twice and eliminating ψxxtt, we can obtain(50)utttt-uxxtt+utt+2ntut+nutt+nttu=0.Equation (49) makes inner product with nttt; then(51)ddtnttt2+nxtt2≤4nttt2+uxxtt22.Furthermore,(52)uxxtt2≤2u¯L∞uttxx+4ux¯L∞uttx+utt2+uxx2+4utxxut¯L∞+4utx2,(53)uttx≤c17utt+α17uttxx,uxxt≤c18uxx+α18uttxx,(54)utttt≤c191-α19uttt,(55)uxxtt≤utttt+utt+nt2+ut2+12n2+12utt2+12u2+12ntt2≤c191-α19uttt+E6+c202E2+α202nttt2+12E2+E5+12C1E0+E6.Due to Lemmas 4 and 6, Corollary 5, and (53) and (54), we have that(56)uxxtt2≤2E51+4E53α18+4E52α17uttxx+4E52c17E6+2E6+4E53c18E6≤B7uttt+B8nttt2+B9,where (57)B7=2E51+4E53α18+4E52α17c191-α19,B8=2E51+4E53α18+4E52α17α202,B9=2E51+4E53α18+4E52α17E6+c202E2+12E2+E5+12C1E0+E6+4E52c17E6+2E6+4E53c18E6.Equation (50) makes inner product with uttt and takes real part; we have that(58)12ddtuttt2≤12uxxtt2+uttt2+12utt2+uttt2+utL∞nt2+uttt2+12nL∞utt2+uttt2+12uL∞ntt+uttt2.Combining (51), (56), and (58) and Corollary 5 and applying Gronwall inequality, Lemma 7 can easily be proven.


## 3. Existence of Local Classical Solution

Firstly, we apply Galerkin method to establish approximating solution for (8)–(11) as follows: (59)nm=nmx,t=∑j=1mnjmtWjx,ϕm=ϕmx,t=∑j=1mφjmtWjx,um=umx,t=∑j=1mujmtWjx,ψm=ψmx,t=∑j=1mujmtWjx.Moreover, nm, ϕm, um, and ψm satisfy the following equations and initial condition:(60)ntm,Wj-ϕxxm,Wj=0,(61)ϕtm,Wj-nm+um2,Wj=0,(62)utm,Wj-ψxxm,Wj=0,(63)ψxxtm,Wj-uxxm,Wj+um,Wj+nmum,Wj=0.(64)nmx,0,Wj=n0,Wj,(65)ϕmx,0,Wj=ϕ0,Wj,(66)umx,0,Wj=u0,Wj,(67)ψmx,0,Wj=ψ0,Wj.Next, by implementing integration estimates for the approximation solution of (60)–(63), we establish local classical solution of (60)–(63).Lemma 8.
Ifum(x,0)∈L2, ψxxm≤C, and um(x,0)=E9, then we have (68)um≤CE9.Proof.
Equation (62) multiplies ujm¯ and makes summation for j=1,2,…,n; we obtain (69)utm,um-ψxxm,um=0.Taking real part, we have that (70)ddtum2≤Cum2.Therefore, by Gronwall inequality, we have um(t)≤CE9.Lemma 9.
Ifnm(x,0)∈L2, nxm(x,0)∈L2, ntm(x,0)∈L2, ϕxm(x,0)∈L2, and umL∞≤E91, then (71)nxm2+ϕm2+ntm2≤E10.Proof.
Equation (60) multiplies njm¯ and makes summation for j=1,2,…,n; we can obtain(72)ntm,nm=12ddtnm2,(73)ϕxxm,nm≤ϕxm2+nxm2.Furthermore, (60) differentiates with respect to t and (61) differentiates with respect to x twice; we have that(74)nttm,wj-nxxm+umxx2,wj=0.Next, (74) multiplies njtm¯ and makes summation for j=1,2,…,n; it is easy to obtain the following equation: (75)nttm,ntm-nxxm+umxx,ntm=0.So, we obtain that(76)nttm,ntm=12ddtntm2,(77)-nxxm,ntm=12ddtnxm2.(78)umxx2,ntm≤12ntm2+4uxm2+uxxmum¯2.Furthermore,(79)uxm2+uxxmum¯≤uxm2+um¯L∞uxxm≤c18um+α19uxxmuxm+c19um¯+α20uxxmuxxm,(80)uxxm≤c201-α21uxm.Therefore,(81)uxm2+uxxmum¯2≤D1+1+D2uxm2,where D1=(c18CE9+c20/1-α21c19CE9)2 and D2=(α19c20/1-α21+α20c202/(1-α21)2)2.
Equation (61) multiplies (-λj); we have that(82)ϕtm,wjxx-nm+um2,wjxx=0.Equation (82) multiplies ϕjm(t); then(83)ϕtm,ϕxxm-nm+um2,ϕxxm=0.Furthermore,(84)ϕtm,ϕxxm=12ddtϕx2,(85)nm+um2,ϕxxm=∫02πnm+um2xϕxdx≤12nxm2+ϕxm2+4umL∞nxm2+uxm¯2≤12nxm2+ϕxm2+4E91nxm2+4uxm2.So,(86)ddtϕxm2≤D3nxm2+ϕxm2+uxm2,where D3=max⁡{4,4E91}.
Combining (76), (77), (78), (81), (84), and (86), we can obtain that(87)ddtnxm2+ntm2+ϕxm2≤D41+ntm2+ntm2+ϕxm2+uxm2,where D4=max⁡{1,2D3+2,8D1,8(1+D2)+2D3}.
So, by Gronwall inequality, we have that(88)nxm2+ϕm2+ntm2≤E10.Lemma 10.
ϕ t m 2 ≤ E 11, ϕxxm≤E12, uttm≤E13, and uxxm≤E14.Proof.
The proof is similar to that of Lemma9. For details, see Lemma 9.Definition 11.
Ifn(x,t)∈L∞(0,T0,H0), ϕ(x,t)∈L∞(0,T0,H2), u(x,t∈L∞(0,T0,H2)), ψ(x,t)∈L∞(0,T0,H3), and nt,ϕt,ut,ψt∈L∞(0,T0,H0). Moreover, they satisfy the following integral identity:(89)nt,v=ϕxx,vϕt,v=n+u2,vut,v=ψxx,vψxxt,v=uxx-u-nu,v,and(90)nx,0,v=n0x,vϕx,0,v=ϕ0x,vux,0,v=u0x,vψx,0,v=ψ0x,v.Then, n, ϕ, u, and ψ are local generalized solutions of (60)–(63). Besides, n, ϕ, u, and ψ are periodic functions with L about x.Theorem 12.
Ifnmx,0∈L2, nxmx,0∈L2, ntmx,0∈L2, ϕxmx,0∈L2, umL∞≤E91, n0, ϕ0, u0, and ψ0 are period functions with L about x. Then, (60)–(63) have local generalized solution.Proof.
By Lemmas8, 9, and 10, nxm, ntm, ϕxm, uxxm, and uttm have uniform boundedness. So, we can select a subsequence and satisfy ntm→weaknt, ϕxm→weakϕx, uttm→weakutt, and uxxm→weakuxx. Moreover, there exists subsequences, nm→strongn, ϕm→strongϕ, and um→strongu.
In (60)–(63), let m→∞; we have(91)nt,Wj-ϕxx,Wj=0,(92)ϕt,Wj-n+u2,Wj=0,(93)ut,Wj-ψxx,Wj=0,(94)ψxxt,Wj-uxx,Wj+u,Wj+nu,Wj=0.
MoreoverWj∈H2 and it is dense in H0,Wj approximates to v. We complete the proof of this theorem. Similarly, we also can prove (64)–(67).Lemma 13.
Ifnttm(x,0), nxtm(x,0), uxxm(x,0), and uttm(x,0) are bound, then(95)nttm2+nxtm2+uxxm2+uttm2≤E10.Proof.
Equation (74) differentiates with respect to t. Next, it multiplies njttm¯ and makes summation for j=1,2,…,n. We can easily obtain the following equation:(96)ntttm,nttm-nxxtm+umxxt2,nttm=0.Furthermore,(97)ntttm,nttm=12ddtnttm2,(98)nxxtm,nttm=-12ddtnxtm2,(99)umxxt,nttm≤12umxxt2+ntt2.Meanwhile, by CoóojieB inequality,(100)umxxt=4utxmuxm¯+2utxxmum¯+2uxxmut¯≤4uxm¯utxmL∞+2uttxmumL∞+2uxxmutmL∞≤4c22um¯+α22uxxm¯c23utm+α23uxxtm+2uttxmc24um¯+α24uxxm¯+2uxxmc25um¯+α25uttm≤D5+D6uttm2+D7uxxm2.Here, (101)D5=4c222C2E92+c271-α27c24CE92+c25CE92,D6=4c232c281-α282+c271-α27+α252,D7=4α222+4α23c261-α262.Moreover, (62) differentiates with respect to t twice, multiplies ujttm¯, and makes summation for j=1,2,…,n; we can obtain that(102)utttm,uttm-ψxxttm,uttm=0,(103)utttm,uttm=12ddtu2,(104)ψxxtt,uttm≤12ψxxttm2+uttm2.
Next, (62) differentiates with respect to x twice, multiplies (-λ), and makes summation for j=1,2,…,n; we have that(105)utxxm,uxxm-ψxxxx,uxxm=0,(106)utxxm,uxxm=12ddtuxxm2,(107)ψxxxx,uxxm≤12uxxm2+ψxxxxm2,
and (60) differentiates with respect to t and x; we can obtain that(108)nttxm,Wj-ϕxxxtm,Wj=0,where (nttxm,Wjx)-(ϕxxxtm,Wjx)=0 multiplies njtm¯ and makes summation for j=1,2,…,n; we can obtain that(109)nttxm,nxt-ϕxxxtm,nxt=0.
Taking real part, we have that(110)nttxm,nxtm=12ddtntxm2,(111)ψxxxtm,nxtm≤12ntxm2+ψxxxtm2.
By (97), (98), (99), (100), (103), (104), (107), (110), and (111) and applying Gronwall inequality, we can obtain the result of Lemma 13.Corollary 14.
Ifutxm(x,0), ψtxm(x,0), and nxxm(x,0) are bound, then there exist T0, 0≤t≤T0, utxm(t)≤E11, ψtxm(t)≤E12, and nxxm(t)≤E13.Theorem 15.
Assume that the conditions of Lemmas8, 9, 10, and 13 are satisfied, then initial value problem (1) has local classical solution.Proof.
The idea and method are similar to Theorem 3.2 of [1]. For details, please see this paper.


## 4. Initial Value Function and Its Derivative Estimate inL2

In this section, we apply the idea and method of [1] to discuss boundedness of ntm(0), ϕtm(0), utm(0), and ψxxtm.Assume thatn0∈H6, ϕ0∈H4, u0∈H4, and ψ0∈H4, and take n0m, ϕ0m, u0m, and ψ0m; they satisfy n0m→H6n0, ϕ0m→H4ϕ0, u0m→H4, and ψ0m→H4ψ0.Due to (64), (65), (66), and (67) and above assumption, it is easy to deduce that n0mH6, ϕ0mH4, u0mH4, and ψ0mH4 have uniform boundedness about m.Next, we denote an orthogonal projection operatorP∈H0, which is spanned by {W1,W2,…,Wm}. By (60), (61), (62), and (63), we have that(112)Pntm0-ϕxxm0=0Pϕtm0-nm0+um02=0Putm0-ψxxm0=0Pψxxtm0-uxxm0+um0+um0nm0=0,and by PY≤Y, we have (113)ntm0≤ϕxxm0,ϕtm0≤ϕtm0+nm0+um02,utm0≤ψxxm,ψxxtm0≤uxxm0+uxxm0+um0+um0nm0.So, we obtain that n0tm, ϕ0tm, u0tm, and ψ0xxtm have uniform boundedness about m.Similarly, it is not difficult to obtain some higher-order derivative estimation, includingn0tm, ϕ0tm, u0tm, and ψ0xxtm. For details, we do not discuss. We have the following Lemma.Lemma 16.
Assume thatn0∈H6, ϕ0∈H4, u0∈H4, ψ0∈H4, then ntm(0), ϕtm(0), utm(0), ψxxtm(0), nttm(0), ϕttm(0), uttm(0), ψxxttm(0), ntttm(0), nttxm(0), ntxxm(0), ϕtttm(0), ϕttxm(0), ϕtxxm(0), uttm(0), umttt(0), utxxm(0), uttxm(0), ψxxttt(0), and ψxxxttm(0) have uniform boundedness about m.


## 5. Existence and Uniqueness of the Global Classical Solution

Theorem 17.
Assume thatn0∈H6, ϕ0∈H4, u0∈H4, ψ0∈H4, and they are periodic functions with L about x; then initial value problem (1) has global classical solution.Proof.
The idea and method are similar to Theorem 4.1 of [1]. For details, please see this paper.Theorem 18.
Global classical solution of (1)-(2) is unique.Proof.
Assume thatu1, n1 and u2, n2 are both classical solutions with the same periodic initial value of (1)-(2). Let V=u1-u2 and W=n1-n2; by (1)-(2), we obtain the following equations about V and W:(114)Vtt-Vxx+V+n1V+Wu2=0,(115)Wtt-Wxx=Vxxu1¯+u2xxV+2Vxu1x¯+u2xVx¯+u1xx¯V+u2Vxx¯,where V is a complex function and W is a real function.
Equation (114) makes inner product with Vt and takes real part; we have that(116)Vtt,Vt=12ddtVt2,(117)Vtx,Vt=-12ddtVx2,(118)n1V,Vt≤12n1L∞V2+Vt2≤12E54V2+Vt2,(119)V,Vt=12ddtV2,(120)Wu2,Vt≤12u2L∞W2+Vt2,≤12E51W2+Vt2.Meanwhile, (115) makes inner product with Wt and takes real part; we have that(121)Wtt,Wt=12ddtWt2,(122)Wxx,Wt=-12ddtWx2,(123)Vxxu1¯,Wt≤12u1¯L∞Vxx2+Wtt2≤12E51c281-α232Vx2+Wt2,(124)u2xxV,Wt≤12u2xxL∞V2+Wt2≤12c28u2+α23u2xxxV2+Wt2≤12c28C1E0+α23E8V2+Wt2,(125)Vxu1x¯,Wt≤12u1xL∞Vx2+Wt2≤12c29u1+α24u1xxVx2+Wt2≤12c29C1E0+α24E6Vx2+Wt2(126)u2xVx¯,Wt≤12u2xL∞Vx2+Wt2≤12c30u2+α25u2xxVx2+Wt2≤12c30C1E0+α25E6Vx2+Wt2,(127)u1xx¯V,Wt≤12u1xx¯L∞V2+Wt2≤12c31u1+α26u1xxxV2+Wt2≤12c31C1E0+α26E8V2+Wt2,(128)u2Vxx¯,Wt≤12u2L∞Vxx2+Wt2≤12E51c281-α232Vx2+Wt2.
Combining (117)–(128), we have that (129)ddtVt2+Vx2+V2+Wt2+Wx2≤D8Vt2+Vx2+V2+Wt2,where(130)D8=max⁡6E54,6E51,12E51c281-α232,6c28C1E0+α23E8,6c29C1E0+α24E6,12c30C1E0+α25E6,6C31C1E0+α26E8.Integrating (130) from [0,T], we obtain that (131)VtT2+VxT2+VT2+WtT2+WxT2e-D8T≤0.Hence, u1=u2 and n1=n2.



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*Source: 4820601-2018-06-28.xml*