# An Initial Boundary Value Problem for the Zakharov Equation

Authors: Quankang Yang; Charles Bu
Journal: International Journal of Partial Differential Equations
 (2013)

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## Abstract

This paper studies an inhomogeneous initial boundary value problem for the one-dimensional Zakharov equation. Existence and uniqueness of the global strong solution are proved by Galerkin’s method and integral estimates.


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## Body

## 1. Introduction

In this paper, we consider the following inhomogeneous initial boundary value problem for the Zakharov equations in one dimension:(1)iEt+Exx=nE,x∈(a,b),t>0,(2)ntt-nxx=(|E|2)xx,x∈(a,b),t>0,(3)E(t,a)=E(t,b)=0,n(t,a)=q1(t),n(t,b)=q2(t),t⩾0,(4)E(0,x)=E0(x),n(0,x)=n0(x),nt(0,x)=n1(x),x∈(a,b).Zakharov equations play an important role in the turbulence theory for plasma waves and resemble closely to the nonlinear Schrödinger equations. There has been extensive study both theoretically and numerically on these equations (e.g., see [1–17]). Most investigation have been focused on the Cauchy problem of the system of Zakharov equations (SZEs), sometimes with a homogeneous boundary condition. It is well known that Zakharov equations possess 1D soliton solutions. Some numerical experiments suggest that the solutions of 2D and 3D SZE may become singular in finite time [7, 18]. Global existence of solutions of the m-dimensional SZE (m⩾2) has been only proved for small initial data [5, 6]. It has been shown that solutions with large initial data may blow up in finite time [8].The system (1)-(2) above describes the interaction of a Langmuir wave and an ion acoustic wave in a plasma (Dendy [19] and Bellan [20]). Here, E(t,x) is an unknown complex vector-valued function that denotes a slowly varying envelope of a highly oscillatory electric field. Meanwhile, n(t,x) is an unknown real function that denotes the fluctuation in the ion density about its equilibrium value (see [19, 21]). We assume that q1(t), q2(t) are given smooth functions.LetQ(t,x) be any C4 function on X=[0,∞)×(a,b) with compact support satisfying Q(t,a)=q1(t),Q(t,b)=q2(t), and we define
(5)V(t,x)=n(t,x)-Q(t,x),G(t,x)=-Qtt+Qxx.
Thus, the problem (1)–(4) is equivalent to
(6)iEt+Exx=(V+Q)E,(7)Vtt-Vxx=(|E|2)xx+G,(8)E(t,a)=E(t,b)=0,V(t,a)=V(t,b)=0,(9)E(0,x)=E0(x),V(0,x)=v0(x),Vt(0,x)=v1(x).In Section2, we obtain the existence of a local weak solution via Galerkin’s method and the principle of compactness. In Section 3, we derive estimates of higher-order derivatives of Galerkin’s approximate solution to obtain the existence and uniqueness of the local strong solution. In Section 3, we prove the existence and uniqueness of the global strong solution.


## 2. Existence of a Local Strong Solution

We first work on Galerkin’s approximation solution for the problem (6)–(9) by choosing the basic functions {wj(x)} as follows:
(10)-wjxx=λjwj,wj|∂Ω=0,wj∈H2(Ω),∥wj∥=1,j=1,2,…N.The approximate solution for the problem (6)–(9) can be written as
(11)EN(t,x)=∑j=1N‍pjN(t)wj,(12)VN(t,x)=∑j=1N‍qjN(t)wj.According to Galerkin’s method, these undetermined coefficientspjN(t),qjN(t) must satisfy the following initial value problem for a system of ordinary differential equations:
(13)i(ENt,wj)+(ENxx,wj)=(ENVN,wj)+(ENQ,wj),(14)(VNtt,wj)-(VNxx,wj)=(|EN|2)xx,wj)+(G,wj)
with pjN(0)=(E0(x),wj), qjN(0)=(v0(x),wj), qjN′(0)=(v1(x),wj) and
(15)EN(0,x)=∑j=1N‍pjN(0)wj,VN(0,x)=∑j=1N‍qjN(0)wj.To obtain existence of a local weak solution, we need the following lemmas.Lemma 1.
Assume thatE0(x)∈L2, then ∥EN(t,x)∥2⩽∥E0(x)∥2.Proof.
We multiply (13) by pjN(t)¯ and sum up in j to obtain
(16)i(ENt,EN)+(ENxx,EN)=(ENVN,EN)+(ENQ,EN).
Since (d/dt)∥EN∥2=2Re(ENt,EN), taing imaginary parts of (16) yields Re(ENt,EN)=0. Therefore, ∥EN∥2 is a constant, and (16) is proved.
From Gagliardo-Nirenberg inequality [22], one has (if u∈H2∩H01)
(17)∥u∥L∞⩽C∥u∥H11/2∥u∥1/2,∥u∥L4⩽C∥ux∥1/4∥u∥3/4,
which will be used in the following estimates.Lemma 2.
LetE0(x)∈H2∩H01, let v0(x)∈H01, let v1(x)∈L2, and let Q(t,x)∈C2(0,T;H2). Then, there is T0>0 such that
(18)∥Φ∥2=∥ENx∥2+∥VNt∥2+∥VNx∥2+∥ENt∥2⩽C(T0),
for any 0⩽t⩽T0, where C is a positive constant depending only on data and T0.Proof.
We multiply (13) by pjN′(t)¯, multiply (14) by qjN′(t), and sum up in j to get
(19)i(ENt,ENt)+(ENxx,ENt)=(ENVN,ENt)+(ENQ,ENt),(20)(VNtt,VNt)-(VNxx,VNt)=((|EN|2)xx,VNt)+(G,VNt).
By taking the imaginary parts of (19), we get
(21)-12ddt∥ENx∥2=12∫Ω‍VNddt|EN|2dx+12∫Ω‍Qddt|EN|2dx.
From (20), we have
(22)12ddt∥VNt∥2+12ddt∥VNx∥2=∫ab‍(|EN|2)xxVNtdx+∫ab‍GVNtdx.
Combining (21) and (22), we find
(23)ddt(∥ENx∥2+∥VNt∥2+∥VNx∥2)=-∫ab‍VNddt|EN|2dx-∫ab‍Qddt|EN|2dx+2∫ab‍(|EN|2)xxVNtdx+2∫ab‍GVNtdx=-2Re∫ab‍ENENt¯VNdx-2Re∫ab‍ENENt¯Qdx+4Re∫ab‍ENENxx¯VNtdx+4∫ab‍|ENx|2VNtdx+2∫ab‍GVNtdx.
We multiply (13) by -λjpjN(t)¯ and sum up j to obtain
(24)∥ENxx∥2=(ENVN,ENxx)+(ENQ,ENxx)-i(ENt,ENxx).
Therefore,
(25)∥ENxx∥⩽∥ENVN+ENQ-iENt∥∥ENxx∥⩽∥ENVN∥+∥ENQ∥+∥ENt∥.
Differentiating (13) in t, we have
(26)i(ENtt,wj)+(ENtxx,wj)=(ENtVN,wj)+(ENVNt,wj)+(ENtQ,wj)+(ENQt,wj),(27)i(ENtt,ENt)+(ENtxx,ENt)=(ENtVN,ENt)+(ENVNt,ENt)+(ENtQ,ENt)+(ENQt,ENt).
By taking the imaginary parts of (27), we obtain the following inequalities:
(28)ddt∥ENt∥2=2Im(ENVNt,ENt)+2Im(ENQt,ENt)ddt∥ENt∥2⩽∥EN∥L∞(∥VNt∥2+2∥ENt∥2+∥Qt∥2).
From (23), we have
(29)ddt(∥ENx∥2+∥VNt∥2+∥VNx∥2)⩽2|∫ab‍ENENt¯VNdx|+2|∫ab‍ENENt¯Qdx|+4|∫ab‍ENENxx¯VNtdx|+4|∫ab‍|ENx|2VNtdx|+2|∫ab‍GVNtdx|⩽2∥EN∥L∞∫ab‍|ENt¯VN|dx+2∥EN∥L∞∫ab‍|ENt¯Q|dx+4∥EN∥L∞∫ab‍|ENxx¯VNt|dx+4∫ab‍|ENx|2|VNt|dx+2∫ab‍|GVNt|dx.
Combining (28), (29), and (25), we get
(30)ddt(∥ENx∥2+∥VNt∥2+∥VNx∥2+∥ENt∥2)⩽∥EN∥L∞(2∫ab‍|ENt¯VN|dx+2∫ab‍|ENt¯Q|dx+4∫ab‍|ENxx¯VNt|dx+∥VNt∥2+2∥ENt∥2+2∥Qt∥2∫ab‍)+4∫ab‍|ENx|2|VNt|dx+∥G∥2+∥VNt∥2⩽∥EN∥L∞(4∥ENt∥2+∥VN∥2+∥Q∥2+2∥ENxx∥2+3∥VNt∥2+2∥Qt∥2)+2∥ENx∥L44+3∥VNt∥2+∥G∥2.
By Gagliardo-Nirenberg’s inequality with m=1, we obtain the following estimates:
(31)∥EN∥L∞⩽C∥EN∥H11/2∥EN∥1/2⩽C∥ENx∥1/2,(32)∥ENVN∥2=∫ab‍|ENVN|2dx⩽12∥EN∥L44+12∥VN∥L44∥ENVN∥2⩽C∥ENx∥∥EN∥3+C∥VNx∥∥VN∥3∥ENVN∥2⩽C∥ENx∥+C∥VNx∥4,(33)∥ENQ∥2⩽C∥ENx∥+C∥Qx∥4.
From (25), (31), (32), and (33), we have
(34)∥ENxx∥2⩽∥ENVN+ENQ-iENt∥2∥ENxx∥2⩽C∥ENx∥+C∥VNx∥4+C∥Qx∥4+3∥ENt∥2,(35)∥ENx∥L44⩽C∥ENxx∥∥ENx∥3⩽C1+∥ENxx∥22∥ENx∥3⩽C∥ENx∥3+C∥ENx∥4+C∥ENx∥6+C∥VNx∥8+32C∥ENtx∥4+C∥Qx∥8
with ∥VN∥⩽C∥VNx∥. Put the above inequalities into (30), and use Young’s inequality a1/pb1/q⩽(a/p)+(b/q) to obtain
(36)ddt∥Φ∥2ddt(∥ENx∥2+∥VNt∥2+∥VNx∥2+∥ENt∥2)⩽C∥ENx∥1/2[4∥ENt∥2+∥VN∥2+∥Q∥2+C∥ENx∥+C∥VNx∥4+C∥Qx∥4+6∥ENt∥2+3∥VNt∥2+2∥Qt∥2]+C∥ENx∥3+C∥ENx∥4+C∥VNx∥6+C∥VNx∥8+3∥ENt∥4+C∥Qx∥8+3∥VNt∥2+∥G∥2⩽C(∥ENx∥2+∥VNt∥2+∥VNx∥2+∥ENt∥2)4+C⩽C∥Φ∥4+C.
Hence, there exist T0>0 such that ∥Φ∥2⩽C(T0) for any 0⩽t⩽T0. Here, C only depends on data and T0 but not N.Lemma 3.
Under the conditions of Lemma2, there exist a positive constant T0, such that
(37)∥ENxx∥2⩽C(T0),∥VN∥2⩽C(T0),∥EN∥L∞2⩽C(T0),∥ENx∥L44⩽C(T0),∥(|EN|2)xx∥2⩽C(T0),
for any 0⩽t⩽T0, and C is a positive constant that depends only on data and T0.Proof.
By Lemma2 and (31), (34), and (35), we have
(38)∥ENxx∥2⩽C(T0),∥EN∥L∞2⩽C(T0),∥ENx∥L44⩽C(T0),∥(|EN|2)xx∥=∥(ENEN¯)xx∥∥(|EN|2)xx∥=∥ENENxx¯+EN¯ENxx+2|EN|2∥∥(|EN|2)xx∥⩽2+∥EN∥L∞2∥ENxx∥2+∥ENx∥L44⩽C(T0).Lemma 4.
Assume thatE0(x)∈H4∩H01, v0(x)∈H2∩H01, v1(x)∈H1, and Q(t,x)∈C2(0,T;H4), then there exist a positive constant T0 such that
(39)∥Ψ∥2=∥ENtx∥2+∥VNtt∥2+∥VNtx∥2+∥ENtt∥2⩽C(T0),∥VNxx∥2⩽C(T0)
for any 0⩽t⩽T0, where C is a positive constant that depends only on data and T0.Proof.
By differentiating (13) twice in t, we have
(40)i(ENttt,wj)+(ENttxx,wj)=((ENVN)tt,wj)+((ENQ)tt,wj).
Multiply (40) by pjN′′(t)¯, and sum up in j to obtain
(41)i(ENttt,ENtt)+(ENttxx,ENtt)=((ENVN)tt,ENtt)+((ENQ)tt,ENtt).
Taking the imaginary parts of (41), we have
(42)12ddt∥ENtt∥2⩽|(ENttVN,ENtt)|+2|(ENtVNt,ENtt)|+|(ENVNtt,ENtt)|+|(ENttQ,ENtt)|+2|(ENtQt,ENtt)|+|(ENQtt,ENtt)|⩽∥VN∥L∞∥ENtt∥2+2∥VNtENtENtt∥L1+∥EN∥L∞∥VNtt∥∥ENtt∥+∥Q∥L∞∥ENtt∥2+2∥Qt∥L∞∥∥ENt∥∥ENtt∥+∥Qtt∥L∞∥EN∥∥ENtt∥.
By Lemmas 2 and 3, we have
(43)∥VN∥L∞⩽C∥VNx∥1/2∥VN∥1/2⩽C(T0),(44)∥VNtENtENtt∥L1⩽∥VNt∥L∞∥ENt∥∥ENtt∥∥VNtENtENtt∥L1⩽C∥VNtx∥1/2∥VNt∥1/2∥ENt∥∥ENtt∥∥VNtENtENtt∥L1⩽C∥VNtx∥1/2∥ENtt∥.
Substituting (43) and (44) into (42), we obtain
(45)12ddt∥ENtt∥2⩽C∥ENtt∥2+C∥VNtx∥1/2∥ENtt∥+C∥VNtt∥∥ENtt∥+C∥ENtt∥2+C∥ENtt∥+C∥ENtt∥⩽C∥ENtt∥2+C∥VNtx∥+C∥VNtt∥2+C.
Differentiating (14) in t, we have
(46)(VNttt,wj)-(VNtxx,wj)=((|EN|2)xxt,wj)+(Gt,wj).
Multiplying (46) by qjN′′(t) and summing up in j, we obtain
(47)(VNtt,VNttt)-(VNtxx,VNtt)=((|EN|2)xxt,VNtt)+(Gt,VNtt).
That is
(48)12ddt∥VNtt∥2+12ddt∥VNtx∥2=((|EN|2)xxt,VNtt)+(Gt,VNtt),
where
(49)|Gt,VNtt|⩽∥Gt∥∥VNtt∥⩽C∥VNtt∥⩽C∥VNtt∥2+C,(50)((|EN|2)xxt,VNtt)=((ENENxx¯+EN¯ENxx+2|ENx|2)t,VNtt)⩽∥ENt∥L∞(∥ENxx∥2+∥VNtt∥2)+∥EN∥L∞(∥ENtxx∥2+∥VNtt∥2)+2∥ENx∥L∞(∥ENtx∥2+∥VNtt∥2)⩽C∥ENtx∥2+C∥VNtt∥4+C∥ENtxx∥2+C.
Differentiating (13) in t, we have
(51)i(ENtt,wj)+(ENtxx,wj)=((ENVN)t,wj)+((ENQ)t,wj).
Multiplying (51) by pjN′′(t)¯ and summing up the above formulas for j form 1 to N, we obtain
(52)i(ENtt,ENtt)+(ENtxx,ENtt)=((ENVN)t,ENtt)+((ENQ)t,ENtt).
Taking the real parts of (52), we get
(53)12ddt∥ENtx∥2⩽∥VN∥L∞(12∥ENt∥2+12∥ENtt∥2)+∥EN∥L∞(12∥VNt∥2+12∥ENtt∥2)+∥Q∥L∞(12∥ENt∥2+12∥ENtt∥2)+∥Qt∥L∞(12∥EN∥2+12∥ENtt∥2)⩽C∥VN∥L∞∥ENtt∥2+C∥VN∥L∞+C∥ENtt∥2+C.
Multiplying (26) by -λjpjN′(t)¯ and summing up in j, we obtain
(54)i(ENtt,ENtxx)+(ENtxx,ENtxx)=((ENVN)t,ENtxx)+((ENQ)t,ENtxx),∥ENtxx∥2=((ENVN)t,ENtxx)+((ENQ)t,ENtxx)-i(ENtt,ENtxx)⩽5(∥ENtVN∥2+∥ENVNt∥2+∥ENtQ∥2+∥ENQt∥2+∥ENtt∥2).
By the above lemmas, we have
(55)∥ENtVN∥2⩽12∥ENt∥L44+12∥VN∥L44∥ENtVN∥2⩽C∥ENtx∥∥ENt∥3+C∥VNx∥∥VN∥3∥ENtVN∥2⩽C∥ENtx∥+C∥VNx∥4⩽C∥ENtx∥+C,(56)∥ENVNt∥2⩽12∥EN∥L44+12∥VNt∥L44∥ENVNt∥2⩽C∥ENx∥∥EN∥3+C∥VNtx∥∥VNt∥∥ENVNt∥2⩽C∥vNtx∥+C,(57)∥ENtQ∥2⩽12∥ENt∥L44+12∥Q∥L44∥ENtQ∥2⩽C∥ENtx∥∥ENt∥3+C⩽C∥ENtx∥+C,(58)∥ENQt∥2⩽∥EN∥L∞2∥Qt∥2⩽C.
Substituting (55)–(58) into (54), we have
(59)∥ENtxx∥2⩽C∥ENtx∥+C∥VNtx∥+5∥ENtt∥+C.
Substituting (59) into (50), we get
(60)((|EN|2)xxt,VNtt)⩽C∥ENtx∥2+C∥VNtt∥4+C(∥ENtx∥+∥VNtx∥+5∥ENtt∥2)+C⩽C∥ENtx∥2+C∥VNtt∥4+C∥VNtx∥+∥ENttx∥2+C.
Substituting (49) and (60) into (48), we obtain
(61)12ddt∥VNtt∥2+12ddt∥VNtx∥2=((|EN|2)xxt,VNtt)+(Gt,VNtt)⩽C∥ENtx∥2+C∥VNtt∥4+C∥VNtx∥+C∥ENttx∥2+C.
Combining (43), (45), (53), and (61), we have
(62)12ddt∥VNtt∥2+12ddt∥VNtx∥2+12ddt∥ENtt∥2+12ddt∥ENtx∥2⩽C(∥ENtx∥2+C∥VNtt∥2+C∥VNtx∥2+C∥ENtt∥2)2+C.
That is
(63)ddt∥Ψ∥2⩽C∥Ψ∥4+C.
Therefore, there exist a positive constantT0, such that
(64)∥Ψ∥2⩽C(T0)
for any 0⩽t⩽T0, and C is a positive constant that depends only on data and T0 (not on N).
Multiplying (14) by -λjqjN(t) and then summing up the above formulas for j form 1 to N, we obtain
(65)(VNtt,VNxx)-(VNxx,VNxx)=((|EN|2)xx,VNxx)+(G,VNxx).
That is
(66)∥VNxx∥2=(VNtt,VNxx)-((|EN|2)xx,VNxx)-(G,VNxx),
where
(67)((|EN|2)xx,VNxx)=2Re∫ab‍ENENxx¯VNxxdx+2∫ab‍|ENx|2VNxxdx⩽2∥EN∥L∞∫ab‍|ENxx¯VNxx|dx+2∫ab‍|ENx|2VNxxdx⩽C∥ENxx∥∥VNxx∥+C∥|ENx|2∥∥VNxx∥.
This implies that(68)∥VNxx∥2⩽∥VNtt∥∥VNxx∥+C∥ENxx∥∥VNxx∥∥VNxx∥2⩽+C∥|ENx|2∥∥VNxx∥+∥G∥∥VNxx∥,∥VNxx∥⩽∥VNtt∥+C∥ENx∥L42+C⩽∥VNtt∥+C∥ENx∥L44+C.
By the above lemmas, we get
(69)∥ENx∥L44⩽C∥ENxx∥∥ENx∥3⩽C.
Hence,
(70)∥VNxx∥⩽C,∥VNxx∥2⩽C.
This completes the proof.Lemma 5 (see [23, Lemma 4]).
LetX⊂E⊂Y be Banach spaces. If the embedding X↪E is compact, then the following embeddings are also compact:(1)
Lq(0,T;X)∩{φ:(∂φ/∂t)∈L1(0,T;Y)}↪Lq(0,T;E),1⩽q⩽∞,(2)
L∞(0,T;X)∩{φ:(∂φ/∂t)∈Lr(0,T;Y)}↪C([0,T];E),1<r⩽∞.Theorem 6.
Under the conditions of Lemma4, then there exists a local strong solution for (6)–(9).Proof.
According to those estimates outlined in the above lemmas, we know that∥ENt∥, ∥VNt∥, ∥ENxx∥, and ∥VNxx∥  are all bounded uniformly in N. By the principle of compactness, there exist subsequences of {ENt}, {VNt}, {ENxx}, and {VNxx} (we still denote these by the same letter) such that
(71)ENt⇀Et,VNt⇀Vt,ENxx⇀Exx,ΔNxx⇀Vxxweakly.
Since∥ENx∥, ∥VNx∥ are uniformly bounded in N, H1(Ω)↪L2(Ω)↪H-1(Ω), we can choose subsequences of {EN}, {VN} (we still denote these by the same index) such that
(72)EN⟶E,VN⟶V,strongly.
Therefore,(73)ENVN⟶EV,stronglyinL∞(0,T0;L2(Ω)).
In fact, we have
(74)∥ENVN-EV∥=∥ENVN-ENV+ENV-EV∥∥ENVN-EV∥⩽∥EN∥L∞∥VN-V∥+∥EN-E∥∥VN∥L∞.
Since(75)∥EN∥L∞⩽C,∥V∥L∞⩽C,∥VN-V∥⟶0,∥EN-E∥⟶0,
we get(76)∥ENVN-EV∥⟶0.
On the other hand, since(77)(|EN|2)xx⇀(|E|2)xx,weaklyinL∞(0,T0;L2(Ω)),
and for all φ∈C0∞(Ω), EN∈L∞(0,T;H2∩H01(Ω)), we see that
(78)((|EN|2)xx-(|E|2)xx,φ)=(|EN|2-|E|2,φxx)=∫ab‍(|EN|-|E|)(|EN|+|E|)φxxdx⩽∥(|EN|+|E|)∥L∞∥φxx∥L∞∥(|EN|-|E|)∥⩽C(∫ab‍|(EN-E)|2dx)1/2=C∥(EN-E)∥⟶0.
AsC0∞(Ω) is dense in L2(Ω), this establishes the local strong solutions and completes the proof of Theorem 6.


## 3. Existence and Uniqueness of a Global Strong Solution

In the following, in order to obtain the global strong solution we will give a priori estimates for the solution which we get in Theorem6.Lemma 7.
Assume thatE0(x)∈L2, then
(79)∥E(t,x)∥2=∥E0(x)∥2,∀t⩾0.Proof.
Multiply both sides of (1) by iE-, integrate the equation on (a,b), and then take the real parts. It is easy to see that (d/dt)∥E∥2=0. This implies that ∥E∥2 is a conserved quantity in respect to time. Therefore, Lemma 7 is proved.Lemma 8.
Assume thatE0(x)∈H2∩H01(Ω), v0(x)∈H01(Ω), v1(x)∈L2(Ω), and Q(t,x)∈C2(0,T;H2(Ω)); then for ∀T>0, t∈[0,T], one has
(80)∥Ex∥2⩽C(T),∥V∥2⩽C(T),∥Vt∥2+∥Et∥2+∥Vx∥2+∥Exx∥2⩽C(T),
where C is a positive constant that depends only on data and T.Proof.
From (7), we have
(81)(V+Q)tt-(V+Q)xx=(|E|2)xx.
Integrating (81) over [0,t], we obtain
(82)(V+Q)t=∫0t‍(V+Q+|E|2)xxdt+n1(x).
Taking the inner product of (6) and Et¯, we get
(83)i(Et,Et)+(Exx,Et)=((V+Q)E,Et).
Taking the real parts of (83), it follows that
(84)-12ddt∥Ex∥2=12∫ab‍(V+Q)(ddt|E|2)dx,ddt∥Ex∥2=-ddt∫ab‍(V+Q)|E|2dx+∫ab‍|E|2(V+Q)tdx.
Substituting (82) into (84), we have
(85)ddt(∥Ex∥2+∫ab‍(V+Q)|E|2dx)=∫ab‍|E|2(V+Q)tdx=∫ab‍|E|2(∫0t‍(V+Q+|E|2)xxdt+n1(x))dx.
By (82), we get
(86)12ddt∫ab‍(V+Q)2dx=∫ab‍(V+Q)(∫0t‍(V+Q+|E|2)xxdt+n1(x))dx.
Integrating by parts yields that(87)12ddt∫ab‍(∫0t‍(V+Q+|E|2)xdt)2dx=-∫ab‍(∫0t‍(V+Q+|E|2)xxdt)(V+Q+|E|2)dx+Q(t,b)(∫0t‍(V+Q+|E|2)xdt)(t,b)-Q(t,a)(∫0t‍(V+Q+|E|2)xdt)(t,a).
Combining (85), (86), and (87), we have
(88)ddt(∥Ex∥2+∫ab‍(V+Q)|E|2dx)+12ddt∫ab‍(V+Q)2dx+12ddt∫ab‍(∫0t‍(V+Q+|E|2)xdt)2dx=∫ab‍(V+Q+|E|2)n1dx+Q(t,b)(∫0t‍(V+Q+|E|2)xdt)(t,b)-Q(t,a)(∫0t‍(V+Q+|E|2)xdt)(t,a).
Integrating (88) over [0,t], we obtain
(89)∥Ex∥2+∫ab‍(V+Q)|E|2dx+12∫ab‍(V+Q)2dx+12∫ab‍(∫0t‍(V+Q+|E|2)xdt)2dx=∥E0x(x)∥2+∫ab‍n0(x)|E0(x)|2dx+12∫ab‍n02(x)dx+∫0t‍∫ab‍(V+Q+|E|2)n1(x)dxdt+F(t),
where
(90)F(t)=∫0t‍[Q(t,b)(∫0t‍(V+Q+|E|2)xdt)(t,b)F(t)=F(t)-Q(t,a)(∫0t‍(V+Q+|E|2)xdt)(t,a)]dtF(t)=∫0t‍∫ab‍ddx[Q(t,x)(∫0t‍(V+Q+|E|2)xdt)]dxdtF(t)=∫0t‍∫ab‍Q(t,x)(∫0t‍(V+Q+|E|2)xxdt)dxdtF(t)=+∫0t‍∫ab‍Qx(t,x)(∫0t‍(V+Q+|E|2)xdt)dxdt.
Substituting (82) into (90), we have
(91)F(t)=∫0t‍∫ab‍Q(t,x)((V+Q)t-n1(x))dxdtF(t)=+∫0t‍∫ab‍Qx(t,x)(∫0t‍(V+Q+|E|2)xdt)dxdtF(t)=∫ab‍[∫0t‍Q(t,x)(V+Q)-Q(0,x)n0(x)F(t)=F(t)-∫0t‍Qt(t,x)(V+Q)dt]dxF(t)=-∫0t‍∫ab‍Q(t,x)n1(x)dxdtF(t)=+∫0t‍∫ab‍Qx(t,x)(∫0t‍(V+Q+|E|2)xdt)dxdt.
Therefore,(92)|F(t)|⩽∫0t‍∫ab‍|Qt(t,x)|(|V+Q+|E|2-|E|2|)dxdt∫ab‍|Q(V+Q+|E|2-|E|2)|dx+∫0t‍∫ab‍|Qx(t,x)|(|∫0t‍(V+Q+|E|2)xdt|)dxdt+C⩽∥Q∥∥V+Q+|E|2∥+∥Q∥L∞∥E∥2+C∫0t‍[∥∫0t‍(V+Q+|E|2)xdt∥2∥V+Q+|E|2∥2+∥E∥2+∥∫0t‍(V+Q+|E|2)xdt∥2]dt+C⩽14∥(V+Q+|E|2)∥2+C∫0t‍[∥V+Q+|E|2∥2+∥∫0t‍(V+Q+|E|2)xdt∥2]dt+C.
Since
(93)(V+Q)|E|2+12(V+Q)2=12(V+Q+|E|2)2-12|E|4,
we have the following estimate after combining (89) and (92):
(94)∥Ex∥2+12∥(V+Q+|E|2)∥2+12∥∫0t‍(V+Q+|E|2)xdt∥2⩽12∥E∥L44+|∫0t‍∫ab‍(V+Q+|E|2)n1(x)dxdt|+|F(t)|⩽Cα2∥Ex∥2+C2α+14∥(V+Q+|E|2)∥2+C∫0t‍[∥∫0t‍(V+Q+|E|2)xdt∥2∥V+Q+|E|2∥2+∥∫0t‍(V+Q+|E|2)xdt∥2]dt+C.
We may chooseα such that Cα<1 to obtain
(95)∥Ex∥2+∥V+Q+|E|2∥2+∥∫0t‍(V+Q+|E|2)xdt∥2⩽C∫0t‍[∥∫0t‍(V+Q+|E|2)xdt∥2∥(V+Q+|E|2)∥2+∥∫0t‍(V+Q+|E|2)xdt∥2]dt+C⩽C∫0t‍[∥∫0t‍(V+Q+|E|2)xdt∥2∥Ex∥2+∥(V+Q+|E|2)∥2+∥∫0t‍(V+Q+|E|2)xdt∥2]dt+C.
By Groonwall’s lemma, we get(96)∥Ex∥2+∥(V+Q+|E|2)∥2+∥∫0t‍(V+Q+|E|2)xdt∥2⩽C(T)
for all t∈[0,T], for any given T>0.
This implies that(97)∥E∥L∞⩽C∥Ex∥1/2∥E∥1/2⩽C(T),(98)∥E∥L4⩽C∥Ex∥1/4∥E∥3/4⩽C(T),(99)∥V∥⩽∥V+Q+|E|2∥+∥Q∥+∥E∥L42⩽C(T),(100)∥V∥2⩽C(T).
Taking the inner product of (6) and Exx¯, we have
(101)i(Et,Exx)+(Exx,Exx)=(EV,Exx)+(EQ,Exx).
By (98), (99), and Lemma 2, we obtain
(102)∥Exx∥⩽∥EV+EQ-iEt∥⩽∥EV∥+∥EQ∥+∥Et∥∥Exx∥⩽∥E∥L∞(∥V∥+∥Q∥)+∥Et∥⩽∥Et∥+C(T).
Taking the inner product of (7) and Vt, we get
(103)(Vtt,Vt)-(Vxx,Vt)=((|E|2)xx,Vt)+(G,Vt),(104)12ddt∥Vt∥2+12ddt∥Vx∥2=∫ab‍(|E|2)xxVtdx+∫ab‍GVtdx.
Differentiating (6) with respect to t then taking the inner product of the result and Et¯, we have
(105)i(Ett,Et)+(Etxx,Et)=(EtV,Et)+(EVt,Et)+(EtQ,Et)+(EQt,Et).
Taking the imaginary parts of (105), we obtain
(106)12ddt∥Et∥2=Im(EVt,Et)+Im(EQt,Et).
Combining (104) and (106), we get
(107)ddt(∥Vt∥2+∥Vx∥2+∥Et∥2)=∫ab‍(|E|2)xxVtdx+∫ab‍GVtdx+Im(EVt,Et)+Im(EQt,Et)=∫ab‍Vt(2ReEExx¯+2|Ex|2)dx+∫ab‍GVtdx+Im(EVt,Et)+Im(EQt,Et)⩽C∥Vt∥(∥Et∥+C)+C∥Exx∥1/2∥Vt∥+C+12∥Vt∥2+C∥Vt∥∥Et∥+C∥Et∥⩽C∥Vt∥2+C∥Et∥2+C⩽C(∥Vt∥2+∥Et∥2+∥Vx∥2)+C.
By Gronwall’s lemma, we obtain
(108)∥Vt∥2+∥Et∥2+∥Vx∥2⩽C(T).
From (95), we get
(109)∥Exx∥⩽∥Et∥+C(T)⩽C(T),∥Exx∥2⩽C(T).
This completes the proof of Lemma8. So, we have the following theorem.Theorem 9.
The solution given by Theorem6 exists for all time, that is, T=∞.The proof of uniqueness is done via standard integral estimates, so, we omit it here.



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*Source: 748761-2013-07-25.xml*