The Asymptotic Solution of the Initial Boundary Value Problem to a Generalized Boussinesq Equation

Abstract

The 𝐿² space solution of an initial boundary problem for a generalized damped Boussinesq equation is constructed. Certain assumptions on the coefficients of the equation are found to show the existence and uniqueness of the global solution to the initial boundary problem. The explicit expression for the large time asymptotic solution is obtained.

1. Introduction

The classical Boussinesq equation can be expressed by 𝑢_𝑡𝑡=−𝛼𝑢_{𝑥𝑥𝑥𝑥}+𝑢_𝑥𝑥+𝛽𝑢²_𝑥𝑥, (1.1) where 𝑢(𝑥,𝑡) is the elevation of the free surface of fluid, the subscripts denote partial derivatives, and the constant coefficients 𝛼 and 𝛽 depend on the depth of fluid and the characteristic speed of the long waves. The Boussinesq equation (1.1) was first derived by Boussinesq [1], in 1872, to describe propagation of long waves with small amplitude on the surface of shallow water. Since then, extensive research has been carried out to study the property and solutions of the equation and its associated initial boundary value problems. Clarkson [2] proposed a general approach for constructing exact solutions of (1.1). Hirota [3] deduced conservation laws and then examined the numerical solution of a Boussinesq equation. Yajima [4] investigated the nonlinear evolution of a linearly stable solution for (1.1). The exponentially decaying solution of the spherical Boussinesq equation has been investigated by Nakamura [5]. Galkin et al. [6] developed rational solutions of the one-dimensional Boussinesq equation with both zero and nonzero boundary conditions at the infinity in the space from the known solutions of the Kadomtzev-Petviashvili equation. The structure of their solutions generalizes a family of rational solutions of the Korteweq-de Vries equation to the case of two-wave processes. For many other methods to investigate the Boussinesq system and other shallow water models, the reader is referred to [7–13] and the references therein.

Various generalizations of the classical Boussinesq equation have been proposed and studied from many aspects, in particular the well-posedness of the Cauchy problem for the following equation, 𝑢_𝑡𝑡=−𝑢_{𝑥𝑥𝑥𝑥}+𝑢_𝑥𝑥+𝑓(𝑢)_𝑥𝑥. (1.2) Bona and Sachs [14] showed that the special solitary wave solutions of (1.2) are nonlinearly stable for a range of wave speeds. The local and global well-posedness of the problem has been proved by transforming (1.2) into a system of nonlinear Schrödinger equations [15].

Varlamov [16] considered the initial-boundary condition for the following damped Boussinesq equation: 𝑢_𝑡𝑡−2𝑏𝑢_𝑡𝑥𝑥=−𝛼𝑢_{𝑥𝑥𝑥𝑥}+𝑢_𝑥𝑥+𝛽𝑢²_𝑥𝑥, (1.3) where the second term on the left-hand side is responsible for dissipation, 𝛽∈𝑅¹, and 𝛼 and 𝑏 are assumed to be positive constants and satisfy the assumption that 𝛼>𝑏². An initial-boundary value problem for (1.3) with small initial data is considered for the case of one space dimension. The classical solution of the problem is constructed, and the long-time asymptotics is obtained in explicit form. The asymptotics show the presence of both time and space oscillations and the exponential decay of the solution in time due to dissipations. Varlamov [17] has also used the eigenfunction expansion method to consider the long-time asymptotics for a damped Boussinesq equation in a ball which is similar to (1.3).

The aim of this paper is to study the initial-boundary value problem for the following generalized damped Boussinesq equation: 𝑢_𝑡𝑡−𝑎𝑢_{𝑡𝑡𝑥𝑥}−2𝑏𝑢_𝑡𝑥𝑥+2𝑏𝑎𝑢_𝑡=−𝑐𝑢_{𝑥𝑥𝑥𝑥}+𝑢_𝑥𝑥−𝑝²𝑢+𝛽𝑢²_𝑥𝑥, (1.4) where 𝑎, 𝑏 and 𝑐 are positive constants and constants 𝑝≥0 and 𝛽∈𝑅¹. We assume the cases 𝑎+𝑐>𝑏² and 𝑝≥√2𝑏/𝑎. The classical solutions of the initial-boundary value problem for (1.4) will be constructed in the form of a Fourier series with coefficients in their own term represented as series in terms of a small parameter involved in the initial conditions. It is shown that the new solution of the initial-boundary value problem for (1.4) is well-posedness. In addition, the long time behavior of our solution also shows the presence of damped oscillations decaying exponentially in time as 𝑡→∞.

2. Theorem of Existence, Uniqueness, and Asymptotics

In this paper, we consider the following initial-boundary value problem for the damped Boussinesq equation: 𝑢_𝑡𝑡−𝑎𝑢_{𝑡𝑡𝑥𝑥}−2𝑏𝑢_𝑡𝑥𝑥+2𝑏𝑎𝑢_𝑡=−𝑐𝑢_{𝑥𝑥𝑥𝑥}+𝑢_𝑥𝑥−𝑝²𝑢+𝛽𝑢²_𝑥𝑥,𝑢()()𝑢0,𝑡=𝑢𝜋,𝑡=0,𝑡>0,_𝑥𝑥(0,𝑡)=𝑢_𝑥𝑥()𝑢()𝜋,𝑡=0,𝑡>0,𝑥,0=𝜀²𝜑(𝑥),𝑢_𝑡(𝑥,0)=𝜀²𝜓(𝑥),𝑥∈(0,𝜋), (2.1) where 𝑎, 𝑏,  𝑐, and 𝑝 are positive constants, 𝛽∈𝑅, and 𝜀 is a small parameter.

Definition 2.1. The function 𝑢(𝑥)∈𝐶^2𝑛(0,𝜋),𝑛≥1, if 𝑢(0)=𝑢(𝜋)=𝑢^′′(0)=𝑢^′′(𝜋)=⋅⋅⋅=𝑢^(2𝑛−2)(0)=𝑢^(2𝑛−2)(𝜋)=0 and 𝑢^(2𝑛)(𝑥)∈𝐿²(0,𝜋).

Definition 2.2. The function 𝑢(𝑥,𝑡) defined on [0,𝜋]×[0,+∞) is said to be the classical solution of the problem defined by (2.1), if it has two continuous derivatives on [0,𝜋]×[0,+∞) and satisfies system (2.1).

The main results obtained are summarized in the following theorem.

Theorem 2.3. If 𝑎>0, 𝑎+𝑐>𝑏², 𝑝≥√2𝑏/𝑎, 𝜑(𝑥)∈𝐶⁸(0,𝜋), 𝜓(𝑥)∈𝐶⁶(0,𝜋), then there is a 𝜀₀>0 such that, for 0<𝜀≤𝜀₀, problem (2.1) has a unique classical solution of which is represented in the form of 𝑢(𝑥,𝑡)=^∞_𝑁=0𝜀^𝑁+1𝑢^(𝑁)(𝑥,𝑡), (2.2) where the function 𝑢^(𝑁)(𝑥,𝑡) is as that defined by expressions (3.24) in the proof below. This series and the series for the derivatives of 𝑢(𝑥,𝑡) involved in (2.1) converge absolutely and uniformly with respect to 𝑥∈[0,𝜋], 𝑡≥0, 𝜀∈[0,𝜀₀]. In addition, the solution of system (2.1) has the following long time behavior as 𝑡→+∞∶𝑢(𝑥,𝑡)=𝑒^{−(𝑏/𝑎)𝑡}(𝐴cos𝜎𝑡+𝐵sin𝜎𝑡)sin𝑥+𝑂e^{[(−𝜂𝑏/𝑎)𝑡]}, (2.3) where 𝜎=𝑎𝑐+(𝑎+𝑐−𝑏²)+(1+𝑎𝑝²−(𝑏²/𝑎²))+𝑝²−(𝑏²/𝑎²)/1+𝑎 is a positive constant, 0<𝜂<1, and the coefficients 𝐴 and 𝐵 are defined by (3.44).

3. Proof of Theorem 2.3

The proof of Theorem 2.3 includes three parts, namely, existence of a solution, uniqueness of the solution, and the asymptotics of the solution. The main techniques used in this paper are based on those presented in [16]. However, it should be emphasized that the method for proving uniqueness of the solution is different from that of [16].

3.1. Existence of a Solution

We make an odd extension of 𝑢(𝑥,𝑡) in 𝑥 to the interval [−𝜋,0] and represent 𝑢 in the form of a complex Fourier series; namely, 𝑢(𝑥,𝑡)=^∞_{𝑛=−∞𝑛≠0}𝑢_𝑛(𝑡)𝑒^𝑖𝑛𝑥,𝑢_𝑛(𝑡)=12𝜋^𝜋_−𝜋𝑢(𝑥,𝑡)𝑒^{−𝑖𝑛𝑥}𝑑𝑥, (3.1) where 𝑢_−𝑛(𝑡)=−𝑢_𝑛(𝑡) for 𝑛≥1. For the above equalities, we shall use the fact that 𝑢(𝑥,𝑡) belongs to the space 𝐿²(−𝜋,𝜋) for each fixed 𝑡>0, and we shall denote the corresponding norm by ‖𝑢(𝑡)‖=‖𝑢(𝑡)‖_{𝐿2(−𝜋,𝜋)}=^𝜋_−𝜋|𝑢(𝑥,𝑡)|²𝑑𝑥^1/2. (3.2) From (3.1), we obtain, through a simple calculation, that 𝑢(𝑥,𝑡)=2𝑖^∞_𝑛=1𝑢_𝑛(𝑡)sin𝑛𝑥,𝑢_𝑛(𝑡)=1𝑖𝜋^𝜋₀𝑢(𝑥,𝑡)sin𝑛𝑥𝑑𝑥. (3.3) Noting the initial functions, we can write on [−𝜋,𝜋] with 𝜑_−𝑛=𝜑_𝑛,𝜓_−𝑛=𝜓_𝑛, (3.4) for 𝑛≥1 that 𝜑(𝑥)=^∞_{𝑛=−∞𝑛≠0}𝜑_𝑛𝑒^𝑖𝑛𝑥,𝜓(𝑥)=^∞_{𝑛=−∞𝑛≠0}𝜓_𝑛𝑒^𝑖𝑛𝑥. (3.5) Then on the interval [0,𝜋], we obtain 𝜑(𝑥)=2𝑖^∞_𝑛=1𝜑_𝑛sin𝑛𝑥,𝜓(𝑥)=2𝑖^∞_𝑛=1𝜓_𝑛𝜑sin𝑛𝑥,_𝑛=1𝑖𝜋^𝜋₀𝜑(𝑥)sin𝑛𝑥𝑑𝑥,𝜓_𝑛=1𝑖𝜋^𝜋₀𝜓(𝑥)sin𝑛𝑥𝑑𝑥. (3.6) Integrating (3.6) by parts and using the smoothness assumption of the initial data, we get the following inequalities: 𝜑_𝑛≤𝐶₁𝑛⁻⁸,𝜓_𝑛≤𝐶₁𝑛⁻⁶,𝑛≥1, (3.7) where 𝐶₁ is a positive constant. Substituting (3.1) and (3.5) into (2.1), we have the following Cauchy problem for the function 𝑢_𝑛(𝑡), 𝑛∈𝑍: 1+𝑎𝑛²𝑢_𝑛^′′(𝑡)+2𝑏𝑛²+𝑏𝑎𝑢^′_𝑛(𝑡)+𝑐𝑛⁴+𝑛²+𝑝²𝑢_𝑛(𝑡)=−𝛽𝑛²𝑃𝑢_𝑛(𝑡),𝑢_𝑛(0)=𝜀²𝜑_𝑛,𝑢^′_𝑛(0)=𝜀²𝜓_𝑛, (3.8) where 𝑃𝑢_𝑛(𝑡)=^∞_{𝑔=−∞𝑔≠0,𝑛}𝑢_𝑛−𝑔(𝑡)𝑢_𝑔(𝑡),𝑢_−𝑛(𝑡)=−𝑢_𝑛(𝑡)for𝑛≥1. (3.9) For 𝑛=1, we have 𝑃𝑢₁(𝑡)=2^∞_𝑔=1𝑢_−𝑔(𝑡)𝑢_1+𝑔(𝑡)=−2^∞_𝑔=1𝑢_𝑔(𝑡)𝑢_1+𝑔(𝑡). (3.10) For 𝑛≥2, it has 𝑃𝑢_𝑛(𝑡)=^𝑛−1_𝑔=1𝑢_𝑛−𝑔(𝑡)𝑢_𝑔(𝑡)−2^∞_𝑔=1𝑢_𝑔(𝑡)𝑢_𝑛+𝑔(𝑡). (3.11) Letting Φ_𝑛=𝜀𝜑_𝑛,Ψ_𝑛=𝜀𝜓_𝑛 and solving the initial value problem (3.8), we get 𝑢_𝑛(𝑡)=𝜀𝑒^{−(𝑏/𝑎)𝑡}cos𝜎_𝑛𝑡+𝑏𝑎⋅sin𝜎_𝑛𝑡𝜎_𝑛Φ_𝑛+sin𝜎_𝑛𝑡𝜎_𝑛Ψ_𝑛−𝛽𝑛²𝜎_𝑛1+𝑎𝑛^2𝑡₀−𝑏exp𝑎(𝑡−𝜏)sin𝜎_𝑛(𝑡−𝜏)𝑃𝑢_𝑛(𝜏)𝑑𝜏, (3.12) where 𝜎_𝑛=𝑎𝑐𝑛⁶+𝑎+𝑐−𝑏²𝑛⁴+1+𝑎𝑝²−(2𝑏/𝑎)𝑛²+𝑝²−𝑏²/𝑎²1+𝑎𝑛²,𝑛≥1,𝑎+𝑐>𝑏²√,𝑝≥2𝑏𝑎. (3.13) Now, in order to solve the integral equation, we use the perturbation theory. Firstly, we express 𝑢_𝑛(𝑡), 𝑛≥1, as a formal series in 𝜀: 𝑢_𝑛(𝑡)=^∞_𝑁=0𝜀^𝑁+1𝜉̂_𝑛^(𝑁)(𝑡). (3.14) Then, by substituting (3.14) into (3.12) and equating the coefficients of the like powers of 𝜀, we get the following formulas, respectively, for 𝑁=0 and 𝑁≥1∶𝜉̂_𝑛⁽⁰⁾(𝑡)=𝜀𝑒^{−(𝑏/𝑎)𝑡}cos𝜎_𝑛𝑡+𝑏𝑎⋅sin𝜎_𝑛𝑡𝜎_𝑛Φ_𝑛+sin𝜎_𝑛𝑡𝜎_𝑛Ψ_𝑛, (3.15)𝜉̂_𝑛^(𝑁)(𝑡)=−𝛽𝑛²𝜎_𝑛1+𝑎𝑛^2𝑡₀−𝑏exp𝑎(𝑡−𝜏)×sin𝜎_𝑛(𝑡−𝜏)𝑄_𝑁𝜉̂_𝑛^(𝑗)(𝜏)𝑑𝜏,𝑁≥1, (3.16) where 𝑛≥1 and 𝑄_𝑁𝜉̂_𝑛^(𝑗)(𝜏)=𝜀_𝑛^𝑛−1𝑁_𝑔=1𝑗=1𝜉̂^(𝑗−1)_𝑛−𝑔(𝜏)𝜉̂_𝑔^{(𝑁−𝑗)}(𝜏)−2^∞𝑁_𝑔=1𝑗=1𝜉̂^(𝑗−1)_𝑛+𝑔(𝜏)𝜉̂_𝑔^{(𝑁−𝑗)}(𝜏), (3.17) in which 𝜀₁=0 and 𝜀_𝑛=1 for 𝑛≥2.

Now we must prove that the formally constructed function (3.3), together with (3.12)–(3.14), does represent a solution of problem (2.1). To do this, we shall show that the series 𝑢(𝑥,𝑡)=^∞_{𝑛=−∞𝑛≠0}𝑒^{∞𝑖𝑛𝑥}_𝑁=0𝜀^𝑁+1𝜉̂_𝑛^(𝑁)(𝑡)=2𝑖^∞_𝑛=1sin𝑛𝑥^∞_𝑁=0𝜀^𝑁+1𝜉̂_𝑛^(𝑁)(𝜏) (3.18) converges absolutely and uniformly. For this purpose, we firstly establish the following time estimates for 𝑛≥1,𝑡>0,𝑁≥0: 𝜉̂_𝑛^(𝑁)(𝑡)≤𝐶^𝑁(𝑁+1)⁻²𝑛⁻⁶𝑒^{−(𝑏/𝑎)𝑡}. (3.19) Here and in the sequel we denote by 𝐶 any positive constant independent of 𝑁, 𝑛, 𝜀, and 𝑡, but possibly depending on the coefficients of the equation and the initial functions.

We shall use the induction on the number 𝑁. For 𝑁=0, we have from (3.7) and (3.15) that 𝜉̂_𝑛⁽⁰⁾(𝑡)≤𝜀𝑒^{−(𝑏/𝑎)𝑡}1+𝑏𝑎𝜎_𝑛Φ_𝑛+1𝜎_𝑛Ψ_𝑛≤𝜀𝑛⁻⁶𝑒^{−(𝑏/𝑎)𝑡}. (3.20) Assuming that (3.19) is valid for all 𝜉̂_𝑛^(𝑠)(𝑡) with 0≤𝑠≤𝑁−1, we shall prove that (3.19) also holds for 𝑠=𝑁. According to [16], for any integer 𝑛≥1, 𝑔≥1, and 𝑔≠𝑛, we have 𝑛−𝑔⁻⁶𝑔⁻⁶≤2⁶𝑛⁻⁶𝑔⁻⁶+𝑛−𝑔⁻⁶,𝑗⁻²𝑁+1−𝑗⁻²≤2²(𝑁+1)⁻²𝑗⁻²+𝑁+1−𝑗⁻². (3.21) From (3.16) and noting that 𝑏𝑛²1+𝑎𝑛²≥𝑏𝑎for𝑛≠0, (3.22) we have 𝜉̂_𝑛^(𝑁)(𝑡)≤𝐶𝛽(𝑁+1)⁻²𝑛^∞−6_𝑔=1𝑔⁻⁶+𝑛−𝑔⁻⁶×^∞_𝑗=1𝐶^𝑗−1𝐶^𝑁−𝑗𝑁+1−𝑗⁻²+𝑗⁻²𝑆_𝑁(𝑛,𝑡),𝑆_𝑁(𝑛,𝑡)≤𝐶𝑒^{−(𝑏/𝑎)𝑡𝑡}₀𝑒^{((𝑏/𝑎)−(2𝑏/𝑎))𝜏}𝑑𝜏≤𝐶𝑒^{−(𝑏/𝑎)𝑡}×𝑒^{((𝑏/𝑎)−(2𝑏/𝑎))𝑡}−1(𝑏/𝑎)−(2𝑏/𝑎)≤𝐶𝑒^{−(𝑏/𝑎)𝑡}. (3.23) Therefore, we know that inequality (3.19) holds.

For the derivation of (3.15), we recall (3.3) and (3.14) with 𝜉̂_𝑛^(𝑁) defined by (3.15) and (3.16) and interchange the order of summations in the series; namely, 𝑢(𝑥,𝑡)=2𝑖^∞_𝑛=1𝑢_𝑛(𝑡)sin𝑛𝑥=2𝑖^∞_𝑛=1sin𝑛𝑥^∞_𝑁=0𝜀^𝑁+1𝜉̂_𝑛^(𝑁)(𝑡)=^∞_𝑁=0𝜀^𝑁+1𝑢^(𝑁)(𝑥,𝑡), (3.24) where 𝑢^(𝑁)(𝑥,𝑡)=2𝑖^∞_𝑛=1𝜉̂_𝑛^(𝑁)sin𝑛𝑥. (3.25) The interchange of the order of summations is allowable as the series is absolutely and uniformly convergent for 𝑥∈[0,𝜋],𝑡≥0 and 𝜀∈[0,𝜀₀]. Differentiating (3.15) and (3.16), we get, for 𝑘=1,2, that 𝜕^𝑡_𝑘𝜉̂_𝑛⁽⁰⁾(𝑡)=^𝑘_𝑙=0𝑐^𝑙_𝑘(−1)^𝑙𝑏𝑎^𝑙𝑒^{−(𝑏/𝑎)𝑡}𝜕_𝑘^𝑘−𝑙cos𝜎_𝑛𝑡+𝑏𝑎sin𝜎_𝑛𝑡𝜎_𝑛Φ_𝑛+sin𝜎_𝑛𝑡𝜎_𝑛Ψ_𝑛,𝜕^𝑘_𝑡𝜉̂_𝑛^(𝑁)(𝑡)=−𝛽𝑛²1+𝑎𝑛²𝜎_𝑛^𝑡₀𝑔_𝑘(𝑛,𝑡−𝜏)𝑄_𝑁𝜉̂_𝑛^(𝑗)(𝜏)𝑑𝜏+𝑅_𝑘(𝑛,𝑡), (3.26) where 𝑔_𝑘(𝑛,𝑡)=^𝑘_𝑙=0𝑐^𝑙_𝑘(−1)^𝑙𝑏𝑎^𝑙𝑒^{−(𝑏/𝑎)𝑡}𝜎_𝑛^𝑘−𝑙sin𝜎_𝑛()𝜋𝑡+𝑘−𝑙2, (3.27)𝑄_𝑁(𝜉̂_𝑛^(𝑗)(𝜏)) is defined by (3.16), 𝑐^𝑙_𝑘(𝑙=0,1,…,𝑘) are binomial coefficients, and 𝑅_𝑘(𝑛,𝑡)(𝑘=1,2) are obtained by differentiating the integral in (3.16) and are as follows: 𝑅₁(𝑛,𝑡)=0,𝑅₂(𝑛,𝑡)=−𝛽𝑛²𝑄_𝑁𝜉̂_𝑛^(𝑗)(𝑡). (3.28) Hence, using (3.19), we deduce that for, 𝑛≥1,𝑁≥0,𝑡>0, and 𝑘≥0,1,2, 𝜕^𝑘_𝑡𝜉̂_𝑛^(𝑁)(𝑡)≤𝐶^𝑁(𝑁+1)⁻²𝑛^2𝑘−6𝑒^{−(𝑏/𝑎)𝑡},𝜕^𝑘_𝑡𝑢_𝑛(𝑡)≤𝐶𝑛^2𝑘−6𝑒^{−(𝑏/𝑎)𝑡}. (3.29) Using these estimates and calculating the necessary derivatives of (3.29), we can prove straightforwardly that (3.29) represents the classical solution of the initial-boundary value problem defined by system (2.1).

3.2. Uniqueness of the Solution

For proving the uniqueness of the constructed solution, we shall assume that there exist two classical solutions 𝑢⁽¹⁾(𝑥,𝑡) and 𝑢⁽²⁾(𝑥,𝑡) to system (2.1) and then deduce that 𝑢⁽¹⁾(𝑥,𝑡) must be equal to 𝑢⁽²⁾(𝑥,𝑡).

Making an odd extension of the two solutions to the segment (−𝜋,0], we notice that both of them belong to the 𝐿²(−𝜋,𝜋) space, and according to Definition 2.2, we have for each fixed time 𝑡>0 that max_{[]𝑥∈−𝜋,𝜋}𝑢⁽¹⁾(𝑥,𝑡)≤𝐶_𝑡,max_{[]𝑥∈−𝜋,𝜋}𝑢⁽²⁾(𝑥,𝑡)<𝐶_𝑡, (3.30) where 𝐶_𝑡 is a constant depending on 𝑡. Let 𝑤(𝑥,𝑡)=𝑢⁽¹⁾(𝑥,𝑡)−𝑢⁽²⁾(𝑥,𝑡), and make an even extension of 𝑤(𝑥,𝑡) to …(−3𝜋,−2𝜋),(−2𝜋,−𝜋),(𝜋,2𝜋),(2𝜋,3𝜋),… Then, we have from (2.1) that 𝑤_𝑡𝑡−𝑎𝑤_{𝑡𝑡𝑥𝑥}−2𝑏𝑤_𝑡𝑥𝑥+2𝑏𝑎𝑤_𝑡=−𝑐𝑤_{𝑥𝑥𝑥𝑥}+𝑤_𝑥𝑥−𝑝²𝑤+𝛽𝑤(𝑥,𝑡)𝑢⁽¹⁾(𝑥,𝑡)+𝑢⁽²⁾(𝑥,𝑡)_𝑥𝑥,𝑤(𝑥,0)=𝑤_𝑡(𝑥,𝑡)=0. (3.31) Taking the Fourier transform of 𝑤 in the interval (−∞,+∞); namely, 𝑤(𝜉,𝑡)=^+∞_−∞𝑒^{−𝑖𝑥𝜉}𝑤(𝑥,𝑡)𝑑𝑥, (3.32) we have 1+𝑎𝜉²𝑤^′′(𝜉,𝑡)+2𝑏𝜉²+𝑏𝑎𝑤(𝜉,𝑡)+𝑐𝜉⁴+𝜉²+𝑝²𝑤(𝜉,𝑡)=−𝛽𝜉²𝑓(𝜉,𝑡), (3.33) where 𝑓(𝑥,𝑡)=𝑤𝑢⁽¹⁾+𝑢⁽²⁾(𝑥,𝑡). (3.34) From (3.33), we have 𝑤(𝜉,𝑡)=−𝛽𝜉²𝜎_𝜉1+𝑎𝜉^2𝑡₀−𝑏exp𝑎(𝑡−𝜏)sin𝜎_𝜉(𝑡−𝜏)𝑓(𝜉,𝜏)𝑑𝜏, (3.35) where 𝜎_𝜉=𝑎𝑐𝜉⁶+(𝑎+𝑐−𝑏)𝜉⁴+1+𝑎𝑝²−2𝑏²/𝑎𝜉²+𝑝²−𝑏²/𝑎²1+𝑎𝜉². (3.36) Thus, it has 𝑤(𝜉,𝑡)≤𝐶^𝑡₀−𝑏exp𝑎(𝑡−𝜏)𝑓(𝜉,𝜏)𝑑𝜏≤𝐶^𝑡₀−𝑏exp𝑎(𝑡−𝜏)𝑑𝜏^1/2𝑡₀𝑓(𝜉,𝜏)²𝑑𝜏^1/2≤𝐶^𝑡₀𝑓(𝜉,𝜏)²𝑑𝜏^1/2. (3.37) It follows from (3.37) and the Parseval inequality that 𝑤(^+∞_−∞𝜉,𝑡)²𝑑𝜉≤𝐶^+∞_−∞^𝑡₀𝑓(𝜉,𝜏)²𝑑𝜏𝑑𝜉≤𝐶^𝑡₀𝑓(𝜉,𝑡)²_𝐿2𝑑𝜏≤𝐶^𝑡₀𝑤(𝑥,𝜏)𝑢⁽¹⁾(𝑥,𝜏)+𝑢⁽²⁾(𝑥,𝜏)²_𝐿2𝑑𝜏≤𝐶^𝑡₀𝐶_𝜏‖𝑤(𝑥,𝜏)‖²_𝐿2𝑑𝜏. (3.38) Using the Gronwall inequality, we obtain 𝑤(𝑥,𝑡)=0in𝐿², (3.39) which implies that 𝑢⁽¹⁾(𝑥,𝑡)≡𝑢⁽²⁾(𝑥,𝑡) and thus the solution is unique.

3.3. Long Time Asymptotics

To find the long-time behavior of the constructed solution, we firstly determine a subtle asymptotic estimate of 𝑢₁(𝑡) which will contribute to the major term and then estimate the remaining terms ∑^∞_𝑛=2𝑢_𝑛(𝑡)sin𝑛𝑥.

Since 𝑢₁(𝑡)=^∞_𝑁=0𝜀^𝑁+1𝜉₁^(𝑁)(𝑡), (3.40) we substitute it into the integral equations (3.15) and (3.16) to obtain 𝜉₁⁽⁰⁾(𝑡)=𝑒^{−(𝑏/𝑎)𝑡}𝐴⁽⁰⁾cos(𝜎𝑡)+𝐵⁽⁰⁾(,𝜉sin𝜎𝑡)_1(𝑁)(𝑡)=𝑒^{−(𝑏/𝑎)𝑡}𝐴^(𝑁)+𝑅_𝐴^(𝑁)(𝑡)cos(𝜎𝑡)+𝐵^(𝑁)+𝑅_𝐵^(𝑁)(𝑡)sin(𝜎𝑡), (3.41) where 𝐴⁽⁰⁾=𝜀𝜑₁,𝐵⁽⁰⁾=𝜀𝜎𝑏𝜑1+𝑎¹+𝜓₁,√𝜎=𝑎𝑐+1+𝑎+𝑐−𝑏²,𝐴1+𝑎_(𝑁)=𝛽𝜎(1+𝑎)^𝑡₀𝑒^{(𝑏/𝑎)𝜏}sin(𝜎𝜏)𝑆_𝑢𝑛(𝜏)𝐵𝑑𝜏,_(𝑁)𝛽=−𝜎(1+𝑎)^𝑡₀𝑒^{(𝑏/𝑎)𝜏}cos(𝜎𝜏)𝑆_𝑢𝑛(𝜏)𝑅𝑑𝜏,_𝐴(𝑁)=𝛽𝜎(1+𝑎)₀^+∞𝑒^{(𝑏/𝑎)𝜏}sin(𝜎𝜏)𝑆_𝑢𝑛(𝜏)𝑅𝑑𝜏,_𝐵(𝑁)𝛽=−𝜎(1+𝑎)₀^+∞𝑒^{(𝑏/𝑎)𝜏}cos(𝜎𝜏)𝑆_𝑢𝑛(𝜏)𝑆𝑑𝜏,_𝑢𝑛(𝑡)=^∞𝑁_𝑔=1𝑗=1𝜉̂^(𝑗−1)_1+𝑔(𝑡)𝜉̂_𝑔^{(𝑁−𝑗)}(𝑡),𝑁≥1, (3.42) and the function 𝜉̂^(𝑗)(𝑡),𝑗=0,1,…,𝑁−1 are defined by (3.15)-(3.16). Taking 𝑛≥2,𝑁≥1 and using a method similar to that used in [16], it follows from (3.19) and (3.24) that there exists a positive number 0<𝜂<1 such that 𝑅_𝐴^(𝑁)(𝑡)≤𝑐𝑒^{−(𝜂𝑏𝑡/𝑎)},𝑅_𝐵^(𝑁)(𝑡)≤𝑐𝑒^{−(𝜂𝑏𝑡/𝑎)}. (3.43) Hence, we have proved that, as 𝑡→+∞, 2𝑖𝑢₁(𝑡)=𝑒^{−(𝑏/𝑎)𝑡}[𝐴cos(𝜎𝑡)+𝐵sin(𝜎𝑡)]+𝑂𝑒^{−((1+𝜂)𝑏/𝑎)𝑡},𝐴=2𝑖^∞_𝑁=0𝜀^𝑁+1𝐴^(𝑁),𝐵=2𝑖^∞_𝑁=0𝜀^𝑁+1𝐵^(𝑁), (3.44) where 0<𝜂<1, 𝐴^(𝑁) and 𝐵^(𝑁) are defined by (3.41), and the series above converge absolutely and uniformly for 𝜀∈[0,𝜀₀].

Now, we can represent the solution by 𝑢(𝑥,𝑡)=2𝑖𝑢₁(𝑡)+𝑅_𝑢(),𝑅𝑥,𝑡_𝑢(𝑥,𝑡)=2𝑖^∞_𝑛=2sin𝑛𝑥^∞_𝑁=0𝜉_𝑛^(𝑁)(𝑡). (3.45) Using (3.24), we deduce that 𝑅_𝑢(𝑥,𝑡)≤exp−𝜂𝑏𝑎𝑡^∞_𝑛=0𝑐^𝑁𝜀^𝑁+1(𝑁+1)^∞−2_𝑁=2𝑛⁻⁶≤𝑐exp−𝜂𝑏𝑎𝑡,𝑢(𝑥,𝑡)=𝑒^{−(𝑏/𝑎)𝑡}(𝐴cos𝜎𝑡+𝐵sin𝜎𝑡)sin𝑥+𝑂𝑒^{−[(𝜂𝑏/𝑎)𝑡]},0<𝜂<1. (3.46)(3.47) It follows from (3.47) that the inequality (2.3) holds.

Acknowledgments

This work is supported by the Fundamental Research Funds for Central Universities (SWUFE 2012).