An Initial Boundary Value Problem for the Zakharov Equation

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.

1. Introduction

In this paper, we consider the following inhomogeneous initial boundary value problem for the Zakharov equations in one dimension: 𝑖𝐸_𝑡+𝐸_𝑥𝑥()𝑛=𝑛𝐸,𝑥∈𝑎,𝑏,𝑡>0,_𝑡𝑡−𝑛_𝑥𝑥=|𝐸|²_𝑥𝑥()𝐸()()(),𝑥∈𝑎,𝑏,𝑡>0,𝑡,𝑎=𝐸𝑡,𝑏=0,𝑛𝑡,𝑎=𝑞₁(𝑡),𝑛(𝑡,𝑏)=𝑞₂(𝑡)𝐸(),𝑡⩾0,0,𝑥=𝐸₀(𝑥),𝑛(0,𝑥)=𝑛₀(𝑥),𝑛_𝑡(0,𝑥)=𝑛₁(𝑥),𝑥∈(𝑎,𝑏). (1)(2)(3)(4)

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.

Let 𝑄(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 𝑉(𝑡,𝑥)=𝑛(𝑡,𝑥)−𝑄(𝑡,𝑥),𝐺(𝑡,𝑥)=−𝑄_𝑡𝑡+𝑄_𝑥𝑥. (5) Thus, the problem (1)–(4) is equivalent to 𝑖𝐸_𝑡+𝐸_𝑥𝑥=()𝑉𝑉+𝑄𝐸,_𝑡𝑡−𝑉_𝑥𝑥=|𝐸|²_𝑥𝑥𝐸()()()()𝐸()+𝐺,𝑡,𝑎=𝐸𝑡,𝑏=0,𝑉𝑡,𝑎=𝑉𝑡,𝑏=0,0,𝑥=𝐸₀(𝑥),𝑉(0,𝑥)=𝑣₀(𝑥),𝑉_𝑡(0,𝑥)=𝑣₁(𝑥). (6)(7)(8)(9)

In Section 2, 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 {𝑤_𝑗(𝑥)} as follows: −𝑤_𝑗𝑥𝑥=𝜆_𝑗𝑤_𝑗,𝑤_𝑗|_𝜕Ω𝑤=0,_𝑗∈𝐻²(Ω),𝑤_𝑗=1,𝑗=1,2,…𝑁. (10)

The approximate solution for the problem (6)–(9) can be written as 𝐸_𝑁(𝑡,𝑥)=^𝑁_𝑗=1𝑝_𝑗𝑁(𝑡)𝑤_𝑗,𝑉_𝑁(𝑡,𝑥)=^𝑁_𝑗=1𝑞_𝑗𝑁(𝑡)𝑤_𝑗. (11)(12)

According to Galerkin’s method, these undetermined coefficients 𝑝_𝑗𝑁(𝑡),𝑞_𝑗𝑁(𝑡) must satisfy the following initial value problem for a system of ordinary differential equations: 𝑖𝐸_𝑁𝑡,𝑤_𝑗+𝐸_𝑁𝑥𝑥,𝑤_𝑗=𝐸_𝑁𝑉_𝑁,𝑤_𝑗+𝐸_𝑁𝑄,𝑤_𝑗𝑉_𝑁𝑡𝑡,,𝑤_𝑗−𝑉_𝑁𝑥𝑥,𝑤_𝑗=𝐸_𝑁²)_𝑥𝑥,𝑤_𝑗+𝐺,𝑤_𝑗 (13)(14) with 𝑝_𝑗𝑁(0)=(𝐸₀(𝑥),𝑤_𝑗), 𝑞_𝑗𝑁(0)=(𝑣₀(𝑥),𝑤_𝑗), 𝑞^′_𝑗𝑁(0)=(𝑣₁(𝑥),𝑤_𝑗) and 𝐸_𝑁(0,𝑥)=^𝑁_𝑗=1𝑝_𝑗𝑁(0)𝑤_𝑗,𝑉_𝑁(0,𝑥)=^𝑁_𝑗=1𝑞_𝑗𝑁(0)𝑤_𝑗. (15)

To obtain existence of a local weak solution, we need the following lemmas.

Lemma 1. Assume that 𝐸₀(𝑥)∈𝐿², then ‖𝐸_𝑁(𝑡,𝑥)‖²⩽‖𝐸₀(𝑥)‖².

Proof. We multiply (13) by 𝑝_𝑗𝑁(𝑡) and sum up in 𝑗 to obtain 𝑖𝐸_𝑁𝑡,𝐸_𝑁+𝐸_𝑁𝑥𝑥,𝐸_𝑁=𝐸_𝑁𝑉_𝑁,𝐸_𝑁+𝐸_𝑁𝑄,𝐸_𝑁. (16) Since (𝑑/𝑑𝑡)‖𝐸_𝑁‖²=2Re(𝐸_𝑁𝑡,𝐸_𝑁), taing imaginary parts of (16) yields Re(𝐸_𝑁𝑡,𝐸_𝑁)=0. Therefore, ‖𝐸_𝑁‖² is a constant, and (16) is proved.
From Gagliardo-Nirenberg inequality [22], one has (if 𝑢∈𝐻²∩𝐻¹₀) ‖𝑢‖_𝐿^∞⩽𝐶‖𝑢‖_𝐻^1/2₁‖𝑢‖^1/2,‖𝑢‖_𝐿4⩽𝐶𝑢_𝑥^1/4‖𝑢‖^3/4, (17) which will be used in the following estimates.

Lemma 2. Let 𝐸₀(𝑥)∈𝐻²∩𝐻¹₀, let 𝑣₀(𝑥)∈𝐻¹₀, let 𝑣₁(𝑥)∈𝐿², and let 𝑄(𝑡,𝑥)∈𝐶²(0,𝑇;𝐻²). Then, there is 𝑇₀>0 such that ‖Φ‖²=𝐸_𝑁𝑥²+𝑉_𝑁𝑡²+𝑉_𝑁𝑥²+𝐸_𝑁𝑡²⩽𝐶𝑇₀, (18) for any 0⩽𝑡⩽𝑇₀, where 𝐶 is a positive constant depending only on data and 𝑇₀.

Proof. We multiply (13) by 𝑝^′_𝑗𝑁(𝑡), multiply (14) by 𝑞^′_𝑗𝑁(𝑡), and sum up in 𝑗 to get 𝑖𝐸_𝑁𝑡,𝐸_𝑁𝑡+𝐸_𝑁𝑥𝑥,𝐸_𝑁𝑡=𝐸_𝑁𝑉_𝑁,𝐸_𝑁𝑡+𝐸_𝑁𝑄,𝐸_𝑁𝑡,𝑉_𝑁𝑡𝑡,𝑉_𝑁𝑡−𝑉_𝑁𝑥𝑥,𝑉_𝑁𝑡=𝐸_𝑁²_𝑥𝑥,𝑉_𝑁𝑡+𝐺,𝑉_𝑁𝑡. (19)(20) By taking the imaginary parts of (19), we get −12𝑑𝐸𝑑𝑡^𝑁𝑥2=12_Ω𝑉_𝑁𝑑𝐸𝑑𝑡^𝑁21𝑑𝑥+2_Ω𝑄𝑑𝐸𝑑𝑡^𝑁2𝑑𝑥. (21) From (20), we have 12𝑑𝑉𝑑𝑡^𝑁𝑡2+12𝑑𝑉𝑑𝑡^𝑁𝑥2=^𝑏_𝑎𝐸_𝑁²_𝑥𝑥𝑉_𝑁𝑡𝑑𝑥+^𝑏_𝑎𝐺𝑉_𝑁𝑡𝑑𝑥. (22) Combining (21) and (22), we find 𝑑𝐸𝑑𝑡^𝑁𝑥2+𝑉_𝑁𝑡²+𝑉_𝑁𝑥²=−^𝑏_𝑎𝑉_𝑁𝑑𝐸𝑑𝑡^𝑁2𝑑𝑥−^𝑏_𝑎𝑄𝑑𝐸𝑑𝑡^𝑁2𝑑𝑥+2^𝑏_𝑎𝐸_𝑁²_𝑥𝑥𝑉_𝑁𝑡𝑑𝑥+2^𝑏_𝑎𝐺𝑉_𝑁𝑡𝑑𝑥=−2Re^𝑏_𝑎𝐸_𝑁𝐸_𝑁𝑡𝑉_𝑁𝑑𝑥−2Re^𝑏_𝑎𝐸_𝑁𝐸_𝑁𝑡𝑄𝑑𝑥+4Re^𝑏_𝑎𝐸_𝑁𝐸_𝑁𝑥𝑥𝑉_𝑁𝑡𝑑𝑥+4^𝑏_𝑎𝐸_𝑁𝑥²𝑉_𝑁𝑡𝑑𝑥+2^𝑏_𝑎𝐺𝑉_𝑁𝑡𝑑𝑥. (23)
We multiply (13) by −𝜆_𝑗𝑝_𝑗𝑁(𝑡) and sum up 𝑗 to obtain 𝐸_𝑁𝑥𝑥²=𝐸_𝑁𝑉_𝑁,𝐸_𝑁𝑥𝑥+𝐸_𝑁𝑄,𝐸_𝑁𝑥𝑥−𝑖𝐸_𝑁𝑡,𝐸_𝑁𝑥𝑥. (24) Therefore, 𝐸_𝑁𝑥𝑥⩽𝐸_𝑁𝑉_𝑁+𝐸_𝑁𝑄−𝑖𝐸_𝑁𝑡⩽𝐸_𝑁𝑉_𝑁+𝐸_𝑁𝑄+𝐸_𝑁𝑡. (25) Differentiating (13) in 𝑡, we have 𝑖𝐸_𝑁𝑡𝑡,𝑤_𝑗+𝐸_{𝑁𝑡𝑥𝑥},𝑤_𝑗=𝐸_𝑁𝑡𝑉_𝑁,𝑤_𝑗+𝐸_𝑁𝑉_𝑁𝑡,𝑤_𝑗+𝐸_𝑁𝑡𝑄,𝑤_𝑗+𝐸_𝑁𝑄_𝑡,𝑤_𝑗,𝑖𝐸_𝑁𝑡𝑡,𝐸_𝑁𝑡+𝐸_{𝑁𝑡𝑥𝑥},𝐸_𝑁𝑡=𝐸_𝑁𝑡𝑉_𝑁,𝐸_𝑁𝑡+𝐸_𝑁𝑉_𝑁𝑡,𝐸_𝑁𝑡+𝐸_𝑁𝑡𝑄,𝐸_𝑁𝑡+𝐸_𝑁𝑄_𝑡,𝐸_𝑁𝑡. (26)(27) By taking the imaginary parts of (27), we obtain the following inequalities: 𝑑𝐸𝑑𝑡^𝑁𝑡2=2Im𝐸_𝑁𝑉_𝑁𝑡,𝐸_𝑁𝑡+2Im𝐸_𝑁𝑄_𝑡,𝐸_𝑁𝑡⩽𝐸_𝑁𝐿∞𝑉_𝑁𝑡²+2𝐸_𝑁𝑡²+𝑄_𝑡². (28) From (23), we have 𝑑𝐸𝑑𝑡^𝑁𝑥2+𝑉_𝑁𝑡²+𝑉_𝑁𝑥²⩽2^𝑏_𝑎𝐸_𝑁𝐸_𝑁𝑡𝑉_𝑁𝑑𝑥+2^𝑏_𝑎𝐸_𝑁𝐸_𝑁𝑡𝑄𝑑𝑥+4^𝑏_𝑎𝐸_𝑁𝐸_𝑁𝑥𝑥𝑉_𝑁𝑡𝑑𝑥+4^𝑏_𝑎𝐸_𝑁𝑥²𝑉_𝑁𝑡𝑑𝑥+2^𝑏_𝑎𝐺𝑉_𝑁𝑡𝐸𝑑𝑥⩽2_𝑁𝐿∞^𝑏_𝑎𝐸_𝑁𝑡𝑉_𝑁𝑑𝑥+2𝐸_𝑁𝐿∞^𝑏_𝑎𝐸_𝑁𝑡𝑄𝐸𝑑𝑥+4_𝑁𝐿∞^𝑏_𝑎𝐸_𝑁𝑥𝑥𝑉_𝑁𝑡𝑑𝑥+4^𝑏_𝑎𝐸_𝑁𝑥²𝑉_𝑁𝑡𝑑𝑥+2^𝑏_𝑎𝐺𝑉_𝑁𝑡𝑑𝑥. (29) Combining (28), (29), and (25), we get 𝑑𝐸𝑑𝑡^𝑁𝑥2+𝑉_𝑁𝑡²+𝑉_𝑁𝑥²+𝐸_𝑁𝑡²⩽𝐸_𝑁𝐿∞2^𝑏_𝑎𝐸_𝑁𝑡𝑉_𝑁𝑑𝑥+2^𝑏_𝑎𝐸_𝑁𝑡𝑄𝑑𝑥+4^𝑏_𝑎𝐸_𝑁𝑥𝑥𝑉_𝑁𝑡𝑑𝑥+𝑉_𝑁𝑡²+2𝐸_𝑁𝑡²+2𝑄_𝑡²+4^𝑏_𝑎𝐸_𝑁𝑥²𝑉_𝑁𝑡𝑑𝑥+‖𝐺‖²+𝑉_𝑁𝑡²⩽𝐸_𝑁𝐿∞4𝐸_𝑁𝑡²+𝑉_𝑁²+‖𝑄‖²+2𝐸_𝑁𝑥𝑥²+3𝑉_𝑁𝑡²+2𝑄_𝑡²+2𝐸_𝑁𝑥⁴_𝐿4+3𝑉_𝑁𝑡²+‖𝐺‖². (30) By Gagliardo-Nirenberg’s inequality with 𝑚=1, we obtain the following estimates: 𝐸_𝑁𝐿∞⩽𝐶𝐸_𝑁𝐻^1/2₁𝐸_𝑁^1/2⩽𝐶𝐸_𝑁𝑥^1/2,𝐸_𝑁𝑉_𝑁²=^𝑏_𝑎𝐸_𝑁𝑉_𝑁²1𝑑𝑥⩽2𝐸_𝑁⁴_𝐿4+12𝑉_𝑁⁴_𝐿4⩽𝐶𝐸_𝑁𝑥𝐸_𝑁³+𝐶𝑉_𝑁𝑥𝑉_𝑁³⩽𝐶𝐸_𝑁𝑥+𝐶𝑉_𝑁𝑥⁴,𝐸_𝑁𝑄²⩽𝐶𝐸_𝑁𝑥+𝐶𝑄_𝑥⁴. (31)(32)(33) From (25), (31), (32), and (33), we have 𝐸_𝑁𝑥𝑥²⩽𝐸_𝑁𝑉_𝑁+𝐸_𝑁𝑄−𝑖𝐸_𝑁𝑡²⩽𝐶𝐸_𝑁𝑥+𝐶𝑉_𝑁𝑥⁴+𝐶𝑄_𝑥⁴+3𝐸_𝑁𝑡²,𝐸_𝑁𝑥⁴_𝐿4⩽𝐶𝐸_𝑁𝑥𝑥𝐸_𝑁𝑥³𝐸⩽𝐶1+_𝑁𝑥𝑥²2𝐸_𝑁𝑥³⩽𝐶𝐸_𝑁𝑥³+𝐶𝐸_𝑁𝑥⁴+𝐶𝐸_𝑁𝑥⁶+𝐶𝑉_𝑁𝑥⁸+32𝐶𝐸_𝑁𝑡𝑥⁴+𝐶𝑄_𝑥⁸ (34)(35) with ‖𝑉_𝑁‖⩽𝐶‖𝑉_𝑁𝑥‖. Put the above inequalities into (30), and use Young’s inequality 𝑎^1/𝑝𝑏^1/𝑞⩽(𝑎/𝑝)+(𝑏/𝑞) to obtain 𝑑‖Φ‖𝑑𝑡²𝑑𝐸𝑑𝑡^𝑁𝑥2+𝑉_𝑁𝑡²+𝑉_𝑁𝑥²+𝐸_𝑁𝑡²⩽𝐶𝐸_𝑁𝑥^1/24𝐸_𝑁𝑡²+𝑉_𝑁²+‖𝑄‖²+𝐶𝐸_𝑁𝑥+𝐶𝑉_𝑁𝑥⁴+𝐶𝑄_𝑥⁴+6𝐸_𝑁𝑡²+3𝑉_𝑁𝑡²+2𝑄_𝑡²+𝐶𝐸_𝑁𝑥³+𝐶𝐸_𝑁𝑥⁴+𝐶𝑉_𝑁𝑥⁶+𝐶𝑉_𝑁𝑥⁸+3𝐸_𝑁𝑡⁴+𝐶𝑄_𝑥⁸+3𝑉_𝑁𝑡²+‖𝐺‖²⩽𝐶𝐸_𝑁𝑥²+𝑉_𝑁𝑡²+𝑉_𝑁𝑥²+𝐸_𝑁𝑡²⁴+𝐶⩽𝐶‖Φ‖⁴+𝐶. (36) 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 Lemma 2, there exist a positive constant 𝑇₀, such that 𝐸_𝑁𝑥𝑥²⩽𝐶𝑇₀,𝑉_𝑁²⩽𝐶𝑇₀,𝐸_𝑁²_𝐿∞⩽𝐶𝑇₀,𝐸_𝑁𝑥⁴_𝐿4⩽𝐶𝑇₀,𝐸_𝑁²_𝑥𝑥²⩽𝐶𝑇₀, (37) for any 0⩽𝑡⩽𝑇₀, and 𝐶 is a positive constant that depends only on data and 𝑇₀.

Proof. By Lemma 2 and (31), (34), and (35), we have 𝐸_𝑁𝑥𝑥²⩽𝐶𝑇₀,𝐸_𝑁²_𝐿∞⩽𝐶𝑇₀,𝐸_𝑁𝑥⁴_𝐿4⩽𝐶𝑇₀,𝐸_𝑁²_𝑥𝑥=𝐸_𝑁𝐸_𝑁𝑥𝑥=𝐸_𝑁𝐸_𝑁𝑥𝑥+𝐸_𝑁𝐸_𝑁𝑥𝑥+2𝐸_𝑁²⩽2+𝐸_𝑁²_𝐿∞𝐸_𝑁𝑥𝑥²+𝐸_𝑁𝑥⁴_𝐿4⩽𝐶𝑇₀. (38)

Lemma 4. Assume that 𝐸₀(𝑥)∈𝐻⁴∩𝐻¹₀, 𝑣₀(𝑥)∈𝐻²∩𝐻¹₀, 𝑣₁(𝑥)∈𝐻¹, and 𝑄(𝑡,𝑥)∈𝐶²(0,𝑇;𝐻⁴), then there exist a positive constant 𝑇₀ such that ‖Ψ‖²=𝐸_𝑁𝑡𝑥²+𝑉_𝑁𝑡𝑡²+𝑉_𝑁𝑡𝑥²+𝐸_𝑁𝑡𝑡²⩽𝐶𝑇₀,𝑉_𝑁𝑥𝑥²⩽𝐶𝑇₀ (39) 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 𝑖𝐸_{𝑁𝑡𝑡𝑡},𝑤_𝑗+𝐸_{𝑁𝑡𝑡𝑥𝑥},𝑤_𝑗=𝐸_𝑁𝑉_𝑁𝑡𝑡,𝑤_𝑗+𝐸_𝑁𝑄_𝑡𝑡,𝑤_𝑗. (40) Multiply (40) by 𝑝^′′_𝑗𝑁(𝑡), and sum up in 𝑗 to obtain 𝑖𝐸_{𝑁𝑡𝑡𝑡},𝐸_𝑁𝑡𝑡+𝐸_{𝑁𝑡𝑡𝑥𝑥},𝐸_𝑁𝑡𝑡=𝐸_𝑁𝑉_𝑁𝑡𝑡,𝐸_𝑁𝑡𝑡+𝐸_𝑁𝑄_𝑡𝑡,𝐸_𝑁𝑡𝑡. (41) Taking the imaginary parts of (41), we have 12𝑑𝐸𝑑𝑡^{𝑁𝑡𝑡2}⩽𝐸_𝑁𝑡𝑡𝑉_𝑁,𝐸_𝑁𝑡𝑡+2𝐸_𝑁𝑡𝑉_𝑁𝑡,𝐸_𝑁𝑡𝑡+𝐸_𝑁𝑉_𝑁𝑡𝑡,𝐸_𝑁𝑡𝑡+𝐸_𝑁𝑡𝑡𝑄,𝐸_𝑁𝑡𝑡+2𝐸_𝑁𝑡𝑄_𝑡,𝐸_𝑁𝑡𝑡+𝐸_𝑁𝑄_𝑡𝑡,𝐸_𝑁𝑡𝑡⩽𝑉_𝑁𝐿∞𝐸_𝑁𝑡𝑡²+2𝑉_𝑁𝑡𝐸_𝑁𝑡𝐸_{𝑁𝑡𝑡𝐿1}+𝐸_𝑁𝐿∞𝑉_𝑁𝑡𝑡𝐸_𝑁𝑡𝑡+‖𝑄‖_𝐿^∞𝐸_𝑁𝑡𝑡²+2𝑄_𝑡𝐿∞𝐸_𝑁𝑡𝐸_𝑁𝑡𝑡+𝑄_{𝑡𝑡𝐿∞}𝐸_𝑁𝐸_𝑁𝑡𝑡. (42) By Lemmas 2 and 3, we have 𝑉_𝑁𝐿∞⩽𝐶𝑉_𝑁𝑥^1/2𝑉_𝑁^1/2⩽𝐶𝑇₀,𝑉_𝑁𝑡𝐸_𝑁𝑡𝐸_{𝑁𝑡𝑡𝐿1}⩽𝑉_{𝑁𝑡𝐿∞}𝐸_𝑁𝑡𝐸_𝑁𝑡𝑡⩽𝐶𝑉_𝑁𝑡𝑥^1/2𝑉_𝑁𝑡^1/2𝐸_𝑁𝑡𝐸_𝑁𝑡𝑡⩽𝐶𝑉_𝑁𝑡𝑥^1/2𝐸_𝑁𝑡𝑡. (43)(44) Substituting (43) and (44) into (42), we obtain 12𝑑𝐸𝑑𝑡^{𝑁𝑡𝑡2}⩽𝐶𝐸_𝑁𝑡𝑡²+𝐶𝑉_𝑁𝑡𝑥^1/2𝐸_𝑁𝑡𝑡+𝐶𝑉_𝑁𝑡𝑡𝐸_𝑁𝑡𝑡+𝐶𝐸_𝑁𝑡𝑡²+𝐶𝐸_𝑁𝑡𝑡+𝐶𝐸_𝑁𝑡𝑡⩽𝐶𝐸_𝑁𝑡𝑡²+𝐶𝑉_𝑁𝑡𝑥+𝐶𝑉_𝑁𝑡𝑡²+𝐶. (45) Differentiating (14) in 𝑡, we have 𝑉_{𝑁𝑡𝑡𝑡},𝑤_𝑗−𝑉_{𝑁𝑡𝑥𝑥},𝑤_𝑗=𝐸_𝑁²_𝑥𝑥𝑡,𝑤_𝑗+𝐺_𝑡,𝑤_𝑗. (46) Multiplying (46) by 𝑞^′′_𝑗𝑁(𝑡) and summing up in 𝑗, we obtain 𝑉_𝑁𝑡𝑡,𝑉_{𝑁𝑡𝑡𝑡}−𝑉_{𝑁𝑡𝑥𝑥},𝑉_𝑁𝑡𝑡=𝐸_𝑁²_𝑥𝑥𝑡,𝑉_𝑁𝑡𝑡+𝐺_𝑡,𝑉_𝑁𝑡𝑡. (47) That is 12𝑑𝑉𝑑𝑡^{𝑁𝑡𝑡2}+12𝑑𝑉𝑑𝑡^{𝑁𝑡𝑥2}=𝐸_𝑁²_𝑥𝑥𝑡,𝑉_𝑁𝑡𝑡+𝐺_𝑡,𝑉_𝑁𝑡𝑡, (48) where 𝐺_𝑡,𝑉_𝑁𝑡𝑡⩽𝐺_𝑡𝑉_𝑁𝑡𝑡⩽𝐶𝑉_𝑁𝑡𝑡⩽𝐶𝑉_𝑁𝑡𝑡²𝐸+𝐶,_{𝑁2𝑥𝑥𝑡},𝑉_𝑁𝑡𝑡=𝐸_𝑁𝐸_𝑁𝑥𝑥+𝐸_𝑁𝐸_𝑁𝑥𝑥+2𝐸_𝑁𝑥²_𝑡,𝑉_𝑁𝑡𝑡⩽𝐸_{𝑁𝑡𝐿∞}𝐸_𝑁𝑥𝑥²+𝑉_𝑁𝑡𝑡²+𝐸_𝑁𝐿∞𝐸_{𝑁𝑡𝑥𝑥}²+𝑉_𝑁𝑡𝑡²+2𝐸_{𝑁𝑥𝐿∞}𝐸_𝑁𝑡𝑥²+𝑉_𝑁𝑡𝑡²⩽𝐶𝐸_𝑁𝑡𝑥²+𝐶𝑉_𝑁𝑡𝑡⁴+𝐶𝐸_{𝑁𝑡𝑥𝑥}²+𝐶. (49)(50) Differentiating (13) in 𝑡, we have 𝑖𝐸_𝑁𝑡𝑡,𝑤_𝑗+𝐸_{𝑁𝑡𝑥𝑥},𝑤_𝑗=𝐸_𝑁𝑉_𝑁𝑡,𝑤_𝑗+𝐸_𝑁𝑄_𝑡,𝑤_𝑗. (51) Multiplying (51) by 𝑝^′′_𝑗𝑁(𝑡) and summing up the above formulas for 𝑗 form 1 to 𝑁, we obtain 𝑖𝐸_𝑁𝑡𝑡,𝐸_𝑁𝑡𝑡+𝐸_{𝑁𝑡𝑥𝑥},𝐸_𝑁𝑡𝑡=𝐸_𝑁𝑉_𝑁𝑡,𝐸_𝑁𝑡𝑡+𝐸_𝑁𝑄_𝑡,𝐸_𝑁𝑡𝑡. (52) Taking the real parts of (52), we get 12𝑑𝐸𝑑𝑡^{𝑁𝑡𝑥2}⩽𝑉_𝑁𝐿∞12𝐸_𝑁𝑡²+12𝐸_𝑁𝑡𝑡²+𝐸_𝑁𝐿∞12𝑉_𝑁𝑡²+12𝐸_𝑁𝑡𝑡²+‖𝑄‖_𝐿^∞12𝐸_𝑁𝑡²+12𝐸_𝑁𝑡𝑡²+𝑄_𝑡𝐿∞12𝐸_𝑁²+12𝐸_𝑁𝑡𝑡²⩽𝐶𝑉_𝑁𝐿∞𝐸_𝑁𝑡𝑡²+𝐶𝑉_𝑁𝐿∞+𝐶𝐸_𝑁𝑡𝑡²+𝐶. (53) Multiplying (26) by −𝜆_𝑗𝑝^′_𝑗𝑁(𝑡) and summing up in 𝑗, we obtain 𝑖𝐸_𝑁𝑡𝑡,𝐸_{𝑁𝑡𝑥𝑥}+𝐸_{𝑁𝑡𝑥𝑥},𝐸_{𝑁𝑡𝑥𝑥}=𝐸_𝑁𝑉_𝑁𝑡,𝐸_{𝑁𝑡𝑥𝑥}+𝐸_𝑁𝑄_𝑡,𝐸_{𝑁𝑡𝑥𝑥},𝐸_{𝑁𝑡𝑥𝑥}²=𝐸_𝑁𝑉_𝑁𝑡,𝐸_{𝑁𝑡𝑥𝑥}+𝐸_𝑁𝑄_𝑡,𝐸_{𝑁𝑡𝑥𝑥}−𝑖𝐸_𝑁𝑡𝑡,𝐸_{𝑁𝑡𝑥𝑥}⩽5𝐸_𝑁𝑡𝑉_𝑁²+𝐸_𝑁𝑉_𝑁𝑡²+𝐸_𝑁𝑡𝑄²+𝐸_𝑁𝑄_𝑡²+𝐸_𝑁𝑡𝑡². (54) By the above lemmas, we have 𝐸_𝑁𝑡𝑉_𝑁²⩽12𝐸_𝑁𝑡⁴_𝐿4+12𝑉_𝑁⁴_𝐿4⩽𝐶𝐸_𝑁𝑡𝑥𝐸_𝑁𝑡³+𝐶𝑉_𝑁𝑥𝑉_𝑁³⩽𝐶𝐸_𝑁𝑡𝑥+𝐶𝑉_𝑁𝑥⁴⩽𝐶𝐸_𝑁𝑡𝑥𝐸+𝐶,_𝑁𝑉_𝑁𝑡²⩽12𝐸_𝑁⁴_𝐿4+12𝑉_𝑁𝑡⁴_𝐿4⩽𝐶𝐸_𝑁𝑥𝐸_𝑁³+𝐶𝑉_𝑁𝑡𝑥𝑉_𝑁𝑡⩽𝐶𝑣_𝑁𝑡𝑥𝐸+𝐶,_𝑁𝑡𝑄²⩽12𝐸_𝑁𝑡⁴_𝐿4+12‖𝑄‖⁴_𝐿4⩽𝐶𝐸_𝑁𝑡𝑥𝐸_𝑁𝑡³+𝐶⩽𝐶𝐸_𝑁𝑡𝑥𝐸+𝐶,_𝑁𝑄_𝑡²⩽𝐸_𝑁²_𝐿∞𝑄_𝑡²⩽𝐶. (55)(56)(57)(58) Substituting (55)–(58) into (54), we have 𝐸_{𝑁𝑡𝑥𝑥}²⩽𝐶𝐸_𝑁𝑡𝑥+𝐶𝑉_𝑁𝑡𝑥+5𝐸_𝑁𝑡𝑡+𝐶. (59) Substituting (59) into (50), we get 𝐸_𝑁²_𝑥𝑥𝑡,𝑉_𝑁𝑡𝑡⩽𝐶𝐸_𝑁𝑡𝑥²+𝐶𝑉_𝑁𝑡𝑡⁴+𝐶𝐸_𝑁𝑡𝑥+𝑉_𝑁𝑡𝑥+5𝐸_𝑁𝑡𝑡²𝐸+𝐶⩽𝐶_𝑁𝑡𝑥²+𝐶𝑉_𝑁𝑡𝑡⁴+𝐶𝑉_𝑁𝑡𝑥+𝐸_{𝑁𝑡𝑡𝑥}²+𝐶. (60) Substituting (49) and (60) into (48), we obtain 12𝑑𝑉𝑑𝑡^{𝑁𝑡𝑡2}+12𝑑𝑉𝑑𝑡^{𝑁𝑡𝑥2}=𝐸_𝑁²_𝑥𝑥𝑡,𝑉_𝑁𝑡𝑡+𝐺_𝑡,𝑉_𝑁𝑡𝑡⩽𝐶𝐸_𝑁𝑡𝑥²+𝐶𝑉_𝑁𝑡𝑡⁴+𝐶𝑉_𝑁𝑡𝑥+𝐶𝐸_{𝑁𝑡𝑡𝑥}²+𝐶. (61)
Combining (43), (45), (53), and (61), we have 12𝑑𝑉𝑑𝑡^{𝑁𝑡𝑡2}+12𝑑𝑉𝑑𝑡^{𝑁𝑡𝑥2}+12𝑑𝐸𝑑𝑡^{𝑁𝑡𝑡2}+12𝑑𝐸𝑑𝑡^{𝑁𝑡𝑥2}⩽𝐶𝐸_𝑁𝑡𝑥²+𝐶𝑉_𝑁𝑡𝑡²+𝐶𝑉_𝑁𝑡𝑥²+𝐶𝐸_𝑁𝑡𝑡²²+𝐶. (62) That is 𝑑‖Ψ‖𝑑𝑡²⩽𝐶‖Ψ‖⁴+𝐶. (63)
Therefore, there exist a positive constant 𝑇₀, such that ‖Ψ‖²⩽𝐶𝑇₀ (64) for any 0⩽𝑡⩽𝑇₀, and 𝐶 is a positive constant that depends only on data and 𝑇₀ (not on 𝑁).
Multiplying (14) by −𝜆_𝑗𝑞_𝑗𝑁(𝑡) and then summing up the above formulas for 𝑗 form 1 to 𝑁, we obtain 𝑉_𝑁𝑡𝑡,𝑉_𝑁𝑥𝑥−𝑉_𝑁𝑥𝑥,𝑉_𝑁𝑥𝑥=𝐸_𝑁²_𝑥𝑥,𝑉_𝑁𝑥𝑥+𝐺,𝑉_𝑁𝑥𝑥. (65) That is 𝑉_𝑁𝑥𝑥²=𝑉_𝑁𝑡𝑡,𝑉_𝑁𝑥𝑥−𝐸_𝑁²_𝑥𝑥,𝑉_𝑁𝑥𝑥−𝐺,𝑉_𝑁𝑥𝑥, (66) where 𝐸_𝑁²_𝑥𝑥,𝑉_𝑁𝑥𝑥=2Re^𝑏_𝑎𝐸_𝑁𝐸_𝑁𝑥𝑥𝑉_𝑁𝑥𝑥𝑑𝑥+2^𝑏_𝑎𝐸_𝑁𝑥²𝑉_𝑁𝑥𝑥𝐸𝑑𝑥⩽2_𝑁𝐿∞^𝑏_𝑎𝐸_𝑁𝑥𝑥𝑉_𝑁𝑥𝑥𝑑𝑥+2^𝑏_𝑎𝐸_𝑁𝑥²𝑉_𝑁𝑥𝑥𝐸𝑑𝑥⩽𝐶_𝑁𝑥𝑥𝑉_𝑁𝑥𝑥+𝐶𝐸_𝑁𝑥²𝑉_𝑁𝑥𝑥. (67)
This implies that 𝑉_𝑁𝑥𝑥²⩽𝑉_𝑁𝑡𝑡𝑉_𝑁𝑥𝑥+𝐶𝐸_𝑁𝑥𝑥𝑉_𝑁𝑥𝑥+𝐶𝐸_𝑁𝑥²𝑉_𝑁𝑥𝑥+‖𝐺‖𝑉_𝑁𝑥𝑥,𝑉_𝑁𝑥𝑥⩽𝑉_𝑁𝑡𝑡+𝐶𝐸_𝑁𝑥²_𝐿4+𝐶⩽𝑉_𝑁𝑡𝑡+𝐶𝐸_𝑁𝑥⁴_𝐿4+𝐶. (68) By the above lemmas, we get 𝐸_𝑁𝑥⁴_𝐿4⩽𝐶𝐸_𝑁𝑥𝑥𝐸_𝑁𝑥³⩽𝐶. (69) Hence, 𝑉_𝑁𝑥𝑥⩽𝐶,𝑉_𝑁𝑥𝑥²⩽𝐶. (70)
This completes the proof.

Lemma 5 (see [23, Lemma 4]). Let 𝑋⊂𝐸⊂𝑌 be Banach spaces. If the embedding 𝑋↪𝐸 is compact, then the following embeddings are also compact:(1)𝐿^𝑞(0,𝑇;𝑋)∩{𝜑∶(𝜕𝜑/𝜕𝑡)∈𝐿¹(0,𝑇;𝑌)}↪𝐿^𝑞(0,𝑇;𝐸),1⩽𝑞⩽∞,(2)𝐿^∞(0,𝑇;𝑋)∩{𝜑∶(𝜕𝜑/𝜕𝑡)∈𝐿^𝑟(0,𝑇;𝑌)}↪𝐶([0,𝑇];𝐸),1<𝑟⩽∞.

Theorem 6. Under the conditions of Lemma 4, then there exists a local strong solution for (6)–(9).

Proof. According to those estimates outlined in the above lemmas, we know that ‖𝐸_𝑁𝑡‖, ‖𝑉_𝑁𝑡‖, ‖𝐸_𝑁𝑥𝑥‖, and ‖𝑉_𝑁𝑥𝑥‖ are all bounded uniformly in 𝑁. By the principle of compactness, there exist subsequences of {𝐸_𝑁𝑡}, {𝑉_𝑁𝑡}, {𝐸_𝑁𝑥𝑥}, and {𝑉_𝑁𝑥𝑥} (we still denote these by the same letter) such that 𝐸_𝑁𝑡⇀𝐸_𝑡,𝑉_𝑁𝑡⇀𝑉_𝑡,𝐸_𝑁𝑥𝑥⇀𝐸_𝑥𝑥,Δ_𝑁𝑥𝑥⇀𝑉_𝑥𝑥weakly. (71)
Since ‖𝐸_𝑁𝑥‖, ‖𝑉_𝑁𝑥‖ are uniformly bounded in 𝑁, 𝐻¹(Ω)↪𝐿²(Ω)↪𝐻⁻¹(Ω), we can choose subsequences of {𝐸_𝑁}, {𝑉_𝑁} (we still denote these by the same index) such that 𝐸_𝑁⟶𝐸,𝑉_𝑁⟶𝑉,strongly. (72)
Therefore, 𝐸_𝑁𝑉_𝑁⟶𝐸𝑉,stronglyin𝐿^∞0,𝑇₀;𝐿²(Ω). (73) In fact, we have 𝐸_𝑁𝑉_𝑁−𝐸𝑉=𝐸_𝑁𝑉_𝑁−𝐸_𝑁𝑉+𝐸_𝑁⩽𝐸𝑉−𝐸𝑉_𝑁𝐿∞𝑉_𝑁−𝑉+𝐸_𝑁−𝐸𝑉_𝑁𝐿∞. (74)
Since 𝐸_𝑁𝐿∞⩽𝐶,‖𝑉‖_𝐿^∞𝑉⩽𝐶,_𝑁−𝑉⟶0,𝐸_𝑁−𝐸⟶0, (75)
we get 𝐸_𝑁𝑉_𝑁−𝐸𝑉⟶0. (76)
On the other hand, since 𝐸_𝑁²_𝑥𝑥⇀|𝐸|²_𝑥𝑥,weaklyin𝐿^∞0,𝑇₀;𝐿²(Ω), (77) and for all 𝜑∈𝐶^∞₀(Ω), 𝐸_𝑁∈𝐿^∞(0,𝑇;𝐻²∩𝐻¹₀(Ω)), we see that 𝐸_𝑁²_𝑥𝑥−|𝐸|²_𝑥𝑥=𝐸,𝜑_𝑁2−|𝐸|²,𝜑_𝑥𝑥=^𝑏_𝑎𝐸_𝑁−|𝐸|𝐸_𝑁+|𝐸|𝜑_𝑥𝑥⩽𝐸𝑑𝑥_𝑁+|𝐸|_𝐿∞𝜑_{𝑥𝑥𝐿∞}𝐸_𝑁−|𝐸|⩽𝐶^𝑏_𝑎𝐸_𝑁−𝐸²𝑑𝑥^1/2=𝐶𝐸_𝑁−𝐸⟶0. (78)
As 𝐶^∞₀(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 Theorem 6.

Lemma 7. Assume that 𝐸₀(𝑥)∈𝐿², then ‖𝐸(𝑡,𝑥)‖²=𝐸₀(𝑥)²,∀𝑡⩾0. (79)

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 that 𝐸₀(𝑥)∈𝐻²∩𝐻¹₀(Ω), 𝑣₀(𝑥)∈𝐻¹₀(Ω), 𝑣₁(𝑥)∈𝐿²(Ω), and 𝑄(𝑡,𝑥)∈𝐶²(0,𝑇;𝐻²(Ω)); then for ∀𝑇>0, 𝑡∈[0,𝑇], one has 𝐸_𝑥²⩽𝐶(𝑇),‖𝑉‖²(𝑇),𝑉⩽𝐶_𝑡2+𝐸_𝑡²+𝑉_𝑥²+𝐸_𝑥𝑥²⩽𝐶(𝑇), (80) where 𝐶 is a positive constant that depends only on data and 𝑇.

Proof. From (7), we have (𝑉+𝑄)_𝑡𝑡−(𝑉+𝑄)_𝑥𝑥=|𝐸|²_𝑥𝑥. (81) Integrating (81) over [0,𝑡], we obtain (𝑉+𝑄)_𝑡=^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑥𝑑𝑡+𝑛₁(𝑥). (82) Taking the inner product of (6) and 𝐸_𝑡, we get 𝑖𝐸_𝑡,𝐸_𝑡+𝐸_𝑥𝑥,𝐸_𝑡=(𝑉+𝑄)𝐸,𝐸_𝑡. (83) Taking the real parts of (83), it follows that −12𝑑𝐸𝑑𝑡^𝑥2=12^𝑏_𝑎()𝑑𝑉+𝑄|𝐸|𝑑𝑡²𝑑𝑑𝑥,𝐸𝑑𝑡^𝑥2𝑑=−𝑑𝑡^𝑏_𝑎(𝑉+𝑄)|𝐸|²𝑑𝑥+^𝑏_𝑎|𝐸|²(𝑉+𝑄)_𝑡𝑑𝑥. (84)
Substituting (82) into (84), we have 𝑑𝐸𝑑𝑡^𝑥2+^𝑏_𝑎(𝑉+𝑄)|𝐸|²=𝑑𝑥_𝑏𝑎|𝐸|²(𝑉+𝑄)_𝑡=𝑑𝑥_𝑏𝑎|𝐸|^2𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑥𝑑𝑡+𝑛₁(𝑥)𝑑𝑥. (85) By (82), we get 12𝑑𝑑𝑡^𝑏_𝑎(𝑉+𝑄)²=𝑑𝑥_𝑏𝑎(𝑉+𝑄)^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑥𝑑𝑡+𝑛₁(𝑥)𝑑𝑥. (86)
Integrating by parts yields that 12𝑑𝑑𝑡^𝑏_𝑎^𝑡₀(𝑉+𝑄+|𝐸|²)_𝑥𝑑𝑡²𝑑𝑥=−^𝑏_𝑎^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑥𝑑𝑡𝑉+𝑄+|𝐸|²()𝑑𝑥+𝑄𝑡,𝑏^𝑡₀𝑉+𝑄+|𝐸|²_𝑥()()𝑑𝑡𝑡,𝑏−𝑄𝑡,𝑎^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡(𝑡,𝑎). (87)
Combining (85), (86), and (87), we have 𝑑𝐸𝑑𝑡^𝑥2+^𝑏_𝑎(𝑉+𝑄)|𝐸|²+1𝑑𝑥2𝑑𝑑𝑡^𝑏_𝑎(𝑉+𝑄)²+1𝑑𝑥2𝑑𝑑𝑡^𝑏_𝑎^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡²=𝑑𝑥_𝑏𝑎𝑉+𝑄+|𝐸|²𝑛₁()𝑑𝑥+𝑄𝑡,𝑏^𝑡₀𝑉+𝑄+|𝐸|²_𝑥)(𝑑𝑡(𝑡,𝑏−𝑄𝑡,𝑎)^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡(𝑡,𝑎). (88)
Integrating (88) over [0,𝑡], we obtain 𝐸_𝑥²+^𝑏_𝑎(𝑉+𝑄)|𝐸|²1𝑑𝑥+2^𝑏_𝑎(𝑉+𝑄)²+1𝑑𝑥2^𝑏_𝑎^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡²=𝐸𝑑𝑥_0𝑥(𝑥)²+^𝑏_𝑎𝑛₀(𝑥)𝐸₀(𝑥)²1𝑑𝑥+2^𝑏_𝑎𝑛²₀(𝑥)+𝑑𝑥_{𝑡0𝑏𝑎}𝑉+𝑄+|𝐸|²𝑛₁(𝑥)𝑑𝑥𝑑𝑡+𝐹(𝑡), (89) where 𝐹(𝑡)=^𝑡₀𝑄(𝑡,𝑏)^𝑡₀𝑉+𝑄+|𝐸|²_𝑥)(𝑑𝑡(𝑡,𝑏−𝑄𝑡,𝑎)^𝑡₀𝑉+𝑄+|𝐸|²_𝑥(=𝑑𝑡𝑡,𝑎)𝑑𝑡_{𝑡0𝑏𝑎}𝑑𝑄()𝑑𝑥𝑡,𝑥^𝑡₀𝑉+𝑄+|𝐸|²_𝑥=𝑑𝑡𝑑𝑥𝑑𝑡_{𝑡0𝑏𝑎}𝑄(𝑡,𝑥)^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑥𝑑𝑡𝑑𝑥𝑑𝑡+^𝑡₀^𝑏_𝑎𝑄_𝑥(𝑡,𝑥)^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡𝑑𝑥𝑑𝑡. (90)
Substituting (82) into (90), we have 𝐹(𝑡)=^𝑡₀^𝑏_𝑎𝑄(𝑡,𝑥)(𝑉+𝑄)_𝑡−𝑛₁(𝑥)+𝑑𝑥𝑑𝑡_{𝑡0𝑏𝑎}𝑄_𝑥(𝑡,𝑥)^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡𝑑𝑥𝑑𝑡=^𝑏_𝑎𝑄(𝑡,𝑥)(𝑉+𝑄)−𝑄(0,𝑥)𝑛₀(𝑥)−^𝑡₀𝑄_𝑡()−𝑡,𝑥)(𝑉+𝑄𝑑𝑡𝑑𝑥_{𝑡0𝑏𝑎}𝑄(𝑡,𝑥)𝑛₁(𝑥)+𝑑𝑥𝑑𝑡_{𝑡0𝑏𝑎}𝑄_𝑥(𝑡,𝑥)^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡𝑑𝑥𝑑𝑡. (91)
Therefore, |𝐹(𝑡)|⩽^𝑡₀^𝑏_𝑎𝑄_𝑡(𝑡,𝑥)𝑉+𝑄+|𝐸|²−|𝐸|²𝑑𝑥𝑑𝑡_𝑏𝑎𝑄𝑉+𝑄+|𝐸|²−|𝐸|²𝑑𝑥+^𝑡₀^𝑏_𝑎𝑄_𝑥(𝑡,𝑥)^𝑡₀𝑉+𝑄+|𝐸|²_𝑥‖𝑄|𝐸|𝑑𝑡𝑑𝑥𝑑𝑡+𝐶⩽‖𝑉+𝑄+²+‖𝑄‖_𝐿^∞‖𝐸‖²+𝐶^𝑡₀𝑉+𝑄+|𝐸|²²+‖𝐸‖²+^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡²1𝑑𝑡+𝐶⩽4𝑉+𝑄+|𝐸|²²+𝐶^𝑡₀𝑉+𝑄+|𝐸|²²+^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡²𝑑𝑡+𝐶. (92) Since (𝑉+𝑄)|𝐸|²+12(𝑉+𝑄)²=12𝑉+𝑄+|𝐸|²²−12|𝐸|⁴, (93) we have the following estimate after combining (89) and (92): 𝐸_𝑥²+12𝑉+𝑄+|𝐸|²²+12^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡²⩽12‖𝐸‖⁴_𝐿4+^𝑡₀^𝑏_𝑎𝑉+𝑄+|𝐸|²𝑛₁(𝑥)+|𝐹(𝑡⩽𝑑𝑥𝑑𝑡)|𝐶𝛼2𝐸_𝑥²+𝐶+12𝛼4𝑉+𝑄+|𝐸|²²+𝐶^𝑡₀𝑉+𝑄+|𝐸|²²+^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡²𝑑𝑡+𝐶. (94)
We may choose 𝛼 such that 𝐶𝛼<1 to obtain 𝐸_𝑥²+𝑉+𝑄+|𝐸|²²+^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡²⩽𝐶^𝑡₀𝑉+𝑄+|𝐸|²²+^𝑡₀(𝑉+𝑄+|𝐸|²)_𝑥𝑑𝑡²𝑑𝑡+𝐶⩽𝐶^𝑡₀𝐸_𝑥²+𝑉+𝑄+|𝐸|²²+^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡²𝑑𝑡+𝐶. (95)
By Groonwall’s lemma, we get 𝐸_𝑥²+𝑉+𝑄+|𝐸|²²+^𝑡₀𝑉+𝑄+|𝐸|²_𝑥𝑑𝑡²⩽𝐶(𝑇) (96) for all 𝑡∈[0,𝑇], for any given 𝑇>0.
This implies that ‖𝐸‖_𝐿^∞⩽𝐶𝐸_𝑥^1/2‖𝐸‖^1/2(𝑇),‖𝐸‖⩽𝐶_𝐿4⩽𝐶𝐸_𝑥^1/4‖𝐸‖^3/4(𝑇),‖𝑉‖⩽|𝐸|⩽𝐶𝑉+𝑄+²+‖𝑄‖+‖𝐸‖²_𝐿4(𝑇)‖𝑉‖⩽𝐶,₂⩽𝐶(𝑇). (97)(98)(99)(100)
Taking the inner product of (6) and 𝐸_𝑥𝑥, we have 𝑖𝐸_𝑡,𝐸_𝑥𝑥+𝐸_𝑥𝑥,𝐸_𝑥𝑥=𝐸𝑉,𝐸_𝑥𝑥+𝐸𝑄,𝐸_𝑥𝑥. (101)
By (98), (99), and Lemma 2, we obtain 𝐸_𝑥𝑥⩽𝐸𝑉+𝐸𝑄−𝑖𝐸_𝑡⩽‖𝐸𝑉‖+‖𝐸𝑄‖+𝐸_𝑡⩽‖𝐸‖_𝐿^∞(‖𝑉‖+‖𝑄‖)+𝐸_𝑡⩽𝐸_𝑡+𝐶(𝑇). (102)
Taking the inner product of (7) and 𝑉_𝑡, we get 𝑉_𝑡𝑡,𝑉_𝑡−𝑉_𝑥𝑥,𝑉_𝑡=|𝐸|²_𝑥𝑥,𝑉_𝑡+𝐺,𝑉_𝑡,12𝑑𝑉𝑑𝑡^𝑡2+12𝑑𝑉𝑑𝑡^𝑥2=^𝑏_𝑎|𝐸|²_𝑥𝑥𝑉_𝑡𝑑𝑥+^𝑏_𝑎𝐺𝑉_𝑡𝑑𝑥. (103)(104)
Differentiating (6) with respect to 𝑡 then taking the inner product of the result and 𝐸_𝑡, we have 𝑖𝐸_𝑡𝑡,𝐸_𝑡+𝐸_𝑡𝑥𝑥,𝐸_𝑡=𝐸_𝑡𝑉,𝐸_𝑡+𝐸𝑉_𝑡,𝐸_𝑡+𝐸_𝑡𝑄,𝐸_𝑡+𝐸𝑄_𝑡,𝐸_𝑡. (105) Taking the imaginary parts of (105), we obtain 12𝑑𝐸𝑑𝑡^𝑡2=Im𝐸𝑉_𝑡,𝐸_𝑡+Im𝐸𝑄_𝑡,𝐸_𝑡. (106) Combining (104) and (106), we get 𝑑𝑉𝑑𝑡^𝑡2+𝑉_𝑥²+𝐸_𝑡²=^𝑏_𝑎|𝐸|²_𝑥𝑥𝑉_𝑡𝑑𝑥+^𝑏_𝑎𝐺𝑉_𝑡𝑑𝑥+Im𝐸𝑉_𝑡,𝐸_𝑡+Im𝐸𝑄_𝑡,𝐸_𝑡=^𝑏_𝑎𝑉_𝑡2Re𝐸𝐸_𝑥𝑥+2𝐸_𝑥²𝑑𝑥+^𝑏_𝑎𝐺𝑉_𝑡𝑑𝑥+Im𝐸𝑉_𝑡,𝐸_𝑡+Im𝐸𝑄_𝑡,𝐸_𝑡⩽𝐶𝑉_𝑡𝐸_𝑡+𝐶+𝐶𝐸_𝑥𝑥^1/2𝑉_𝑡1+𝐶+2𝑉_𝑡²+𝐶𝑉_𝑡𝐸_𝑡+𝐶𝐸_𝑡⩽𝐶𝑉_𝑡²+𝐶𝐸_𝑡²𝑉+𝐶⩽𝐶_𝑡²+𝐸_𝑡²+𝑉_𝑥²+𝐶. (107) By Gronwall’s lemma, we obtain 𝑉_𝑡²+𝐸_𝑡²+𝑉_𝑥²⩽𝐶(𝑇). (108) From (95), we get 𝐸_𝑥𝑥⩽𝐸_𝑡(𝑇)(𝑇),𝐸+𝐶⩽𝐶_𝑥𝑥2⩽𝐶(𝑇). (109)
This completes the proof of Lemma 8. So, we have the following theorem.

Theorem 9. The solution given by Theorem 6 exists for all time, that is, 𝑇=∞.

The proof of uniqueness is done via standard integral estimates, so, we omit it here.

Acknowledgments

This work is supported by Qing Lan and “333” Project of Jiangsu Province and the NSF of the Jiangsu Higher Education Committee of China (11KJA110001). It is supported by the Brachman Hoffman Small Grant and Wellesley College Faculty Award.