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

and Applied Analysis 3 2. Theorem of Existence, Uniqueness, and Asymptotics In this paper, we consider the following initial-boundary value problem for the damped Boussinesq equation: utt − auttxx − 2butxx 2b a ut −cuxxxx uxx − p2u β ( u2 ) xx , u 0, t u π, t 0, t > 0, uxx 0, t uxx π, t 0, t > 0, u x, 0 ε2φ x , ut 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 ∈ C2n 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 0 ε 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 [ A cos σt B sin σt sinx O ( e −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 . 4 Abstract and Applied Analysis 3.1. Existence of a Solution Wemake 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 ûn t e, ûn t 1 2π ∫π −π u x, t e−inxdx, 3.1 where û−n t −ûn 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 |dx )1/2 . 3.2 From 3.1 , we obtain, through a simple calculation, that u x, t 2i ∞ ∑ n 1 ûn t sinnx, ûn t 1 iπ ∫π 0 u x, t sinnx dx. 3.3 Noting the initial functions, we can write on −π,π with φ̂−n φ̂n, ψ̂−n ψ̂n, 3.4


Introduction
The classical Boussinesq equation can be expressed by 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 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 Schr ödinger equations 15 .
Varlamov 16 considered the initial-boundary condition for the following damped Boussinesq equation: where the second term on the left-hand side is responsible for dissipation, β ∈ R 1 , and α and b are assumed to be positive constants and satisfy the assumption that α > b 2 .An initialboundary 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: where a, b and c are positive constants and constants p ≥ 0 and β ∈ R 1 .We assume the cases a c > b 2 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 wellposedness.In addition, the long time behavior of our solution also shows the presence of damped oscillations decaying exponentially in time as t → ∞.

Theorem of Existence, Uniqueness, and Asymptotics
In this paper, we consider the following initial-boundary value problem for the damped Boussinesq equation: where a, b, c, and p are positive constants, β ∈ R, and ε is a small parameter.
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.
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, ε 0 .In addition, the solution of system 2.1 has the following long time behavior as t → ∞ : and the coefficients A and B are defined by 3.44 .

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 .

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, 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 L 2 −π, π for each fixed t > 0, and we shall denote the corresponding norm by

3.2
From 3.1 , we obtain, through a simple calculation, that Noting the initial functions, we can write on −π, π with ψ n e inx .

3.5
Then on the interval 0, π , we obtain

3.6
Integrating 3.6 by parts and using the smoothness assumption of the initial data, we get the following inequalities: where C 1 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: 3.8 where 3.9 For n 1, we have For n ≥ 2, it has u g t u n g t .

3.11
Letting Φ n ε ϕ n , Ψ n ε ψ n and solving the initial value problem 3.8 , we get where

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: 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 : where n ≥ 1 and .17 in which ε 1 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 converges absolutely and uniformly.For this purpose, we firstly establish the following time estimates for n ≥ 1, t > 0, N ≥ 0:

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

3.20
Abstract and Applied Analysis 7 Assuming that 3.19 is valid for all ξ s n 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

3.21
From 3.16 and noting that for n / 0, 3.22

3.23
Therefore, we know that inequality 3.19 holds.For the derivation of 3.15 , we recall 3.3 and 3.14 with ξ N n defined by 3.15 and 3.16 and interchange the order of summations in the series; namely, sin nx.

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, ε 0 .Differentiating 3.15 and 3.16 , we get, for k 1, 2, that

3.26
where n 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:

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 .

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 L 2 −π, π space, and according to Definition 2.

3.37
It follows from 3.37 and the Parseval inequality that

3.38
Using the Gronwall inequality, we obtain w x, t 0 in L 2 , 3.39 which implies that u 1 x, t ≡ u 2 x, t and thus the solution is unique.

Long Time Asymptotics
To find the long-time behavior of the constructed solution, we firstly determine a subtle asymptotic estimate of u 1 t which will contribute to the major term and then estimate the remaining terms ∞ n 2 u n t sin nx.Since we substitute it into the integral equations 3.15 and 3.16 to obtain t sin σt ,

3.41
where and the function ξ 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

3.43
Hence, we have proved that, as t → ∞, 2i u 1 t e − b/a t A cos σt B sin σt O e − 1 η b/a t , A 2i

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, ε 0 .
− b/a t A cos σt B sin σt sin x O e − ηb/a t , 0 < η < 1.3.47Itfollows from 3.47 that the inequality 2.3 holds.