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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 364165, 7 pages
http://dx.doi.org/10.1155/2013/364165
Research Article

Global Existence and Asymptotic Behavior of Solutions to the Generalized Damped Boussinesq Equation

1School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
2College of Science, Henan Institute of Engineering, Zhengzhou 450001, China

Received 26 January 2013; Accepted 12 July 2013

Academic Editor: B. G. Konopelchenko

Copyright © 2013 Yinxia Wang and Hengjun Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the Cauchy problem for the generalized damped Boussinesq equation. Under small condition on the initial value, we prove the global existence and optimal decay estimate of solutions for all space dimensions . Moreover, when , we show that the solution can be approximated by the linear solution as time tends to infinity.

1. Introduction

We investigate the Cauchy problem of the following generalized damped Boussinesq equation: with the initial value Here is the unknown function of and , , and are constants. The nonlinear term is a given smooth function of satisfying for .

It is well known that the classical Boussinesq equation was derived by Boussinesq [1] in 1872 to describe shallow water waves, where is an elevation of the free surface of fluid and the constant coefficients and depend on the depth of fluid and the characteristic speed of long waves. It is interesting to note that this equation also governs nonlinear string oscillations.

Taking into account dispersion and nonlinearity, but in real processes viscosity also plays an important role. Varlamov considered the following damped Boussinesq equation (see [24]): where and are constants. Under small condition on the initial value, Varlamov [2] obtained a classical solution to the problem (4), (2) by means of the application of both the spectral and perturbation theories. Moreover, large time asymptotics of this solution was also discussed. For the problem (4), (2) in one, two, and three space dimensions, existence and uniqueness of local solution are proved by Varlamov [3]. The author also showed that for discontinuous initial perturbations this solution is infinitely differentiable with respect to time and space coordinates for on a bounded time interval. Existence and uniqueness of the classical solution for the problem (4), (2) in two space dimensions was proved, and the solution was constructed in the form of a series. The major term of its long-time asymptotics is calculated explicitly, and a uniform in space estimate of the residual term was given (see [4]).

The main purpose of this paper is to establish the following optimal decay estimate of solutions to (1), (2) for : for and . Here is assumed to be small. Moreover, when , we show that our solution can be approximated by the solution to the linearized problem, namely, the problem (1), (2) with . More precisely, when , we show that for and , where for and for .

The study of the global existence and asymptotic behavior of solutions to hyperbolic-type equations has a long history. We refer to [5, 6] for hyperbolic equations, [710] for damped wave equation and [1117] for various aspects of dissipation of the plate equation.

The paper is organized as follows. In Section 2, we derive the solution formula of our semilinear problem. We study the decay property of the solution operators appearing in the solution formula in Section 3. Then, in Section 4, we prove the global existence and asymptotic decay of solutions. Finally, we derive a simpler asymptotic profile which gives the approximation to the linear solution in Section 5.

Notations. We give some notations which are used in this paper. Let denote the Fourier transform of defined by

For , denotes the usual Lebesgue space with the norm . The usual Sobolev space of is defined by with the norm ; the homogeneous Sobolev space of is defined by with the norm ; especially , . Moreover, we know that for .

Finally, in this paper, we denote every positive constant by the same symbol or without confusion. is the Gauss symbol.

2. Solution Formula

The aim of this section is to derive the solution formula for the problem (1), (2). We first investigate the linearized equation of (1): with the initial data in (2). We apply the Fourier transform to (8). This yields The corresponding initial value is given as The characteristic equation of (9) is Let be the corresponding eigenvalues of (11), and we obtain The solution to the problem (9), (10) in the Fourier space is then given explicitly in the form where We define and by respectively, where denotes the inverse Fourier transform. Then, applying to (13), we obtain By the Duhamel principle, we obtain the solution formula to (1), (2) as follows:

3. Decay Property

The aim of this section is to establish decay estimates of the solution operators and appearing in (16) and (17), respectively.

Lemma 1. The solution of the problem (9), (10) satisfies for and .

Proof. Multiplying (9) by and taking the real part yield Multiplying (9) by and taking the real part, we obtain Multiplying both sides of (22) by and summing up the resulting equation and (21) yield where A simple computation implies that where Note that It follows from (25) that Using (23) and (28), we get Thus which together with (25) proves the desired estimates (20). Then we have completed the proof of the lemma.

Lemma 2. Let and be the fundamental solution of (8) in the Fourier space, which are given in (14) and (15), respectively. Then one has the estimates for and .

Proof. Firstly, we investigate the problem (8), (2) with ; from (13), we obtain Substituting the equalities into (20) with , we get (31).
In what follows, we consider the problem (8), (2) with ; it follows from (13) that Substituting the equalities into (20) with , we get the desired estimate (32). The lemma is proved.

Lemma 3. Let and be the fundamental solution of (8), which are given in (16) and (17), respectively. Let , and let , , and be nonnegative integers. Then one has for , where in (35). Similarly, one has for .

Proof. We only prove (35). By the Plancherel theorem, (31), and Hausdorff-Young inequality, we obtain For the term , letting , we have where we used the Hölder inequality with and the Hausdorff-Young inequality for . On the other hand, we can estimate the term simply as where .
Combining the previous three inequalities yields (35). This completes the proof of Lemma 3.

From Lemma 3, we immediately have the following corollary.

Corollary 4. Let and be the fundamental solution of (8), which are given in (15) and (16), respectively. Let , and let , , and be nonnegative integers. Then one has for . Also one has for .

4. Global Existence and Asymptotic Behavior of Solutions to (1), (2)

The purpose of this section is to prove global existence and asymptotic behavior of solutions to the Cauchy problem (1), (2). We need the following lemma, which come from [18] (see also [19]).

Lemma 5. Assume that is a smooth function. Suppose that ( is an integer) when . Then for integer , if and , then the following inequalities hold: where , , .

Theorem 6. Let and . Suppose that is smooth and satisfies for . Assume that , . Put Then there exists a positive constant such that if , the Cauchy problem (1), (2) has a unique global solution satisfying Moreover, the solution satisfies the decay estimate where in (47) and in (48).

Proof. The existence and uniqueness of small solutions can be proved by the contraction mapping principle. Here we only show the decay estimates (47) and (48) for the solution of (19) satisfying with some . To this end, we introduce the quantity Here we note that provided that . This follows from the Gagliardo-Nirenberg inequality with and and the definition of in (49).
Applying to (19) and taking the norm, we obtain Firstly, we estimate . We apply (35) with , , and for , for ). This yields where . Similarly, applying (36) with , , and to the term , we have We estimate the nonlinear term . We divide into two parts and write , where and are corresponding to the time intervals and , respectively. For the term , we apply (42) with , , and . This yields Here we see that by Lemma 5. Thus we have . Therefore we can estimate the term as where On the other hand, we have by Lemma 5. Therefore, using (50), we find that . Consequently, we can estimate the term as Finally, we estimate the term on the time interval . Applying (42) with , , and and using , we obtain Thus we have shown that Substituting all these estimates into (51), we obtain for . Consequently, we have , from which we can deduce , provided that is suitably small. This proves the decay estimate (47).
In what follows, we prove the decay estimate (48) for the time derivative . For this purpose, we differentiate (19) with respect to to obtain Applying to (61) and taking the norm, we have where . For the term , we apply (37) with , , and to get Also, for the term , applying (38) with , , and , we have To estimate the nonlinear term , we divide as , where and correspond to the time intervals and , respectively. For the term , we apply (43) with , , and . This yields Since as before, we can estimate the term as Also, the term is estimated similarly as before, and we have . Finally, we estimate the term by applying (43) with , , and and obtain where we used the estimate . Consequently we have shown that Substituting all these estimates together with the previous estimate into (62), we arrive at the desired estimate (48) for . This completes the proof of Theorem 6.

The previous proof of Theorem 6 shows that when , the solution to the integral equation (19) is asymptotic to the linear solution given by the formula in (18) as . This result is stated as follows.

Corollary 7. Let and assume the same conditions of Theorem 6. Then the solution of the problem (1), (2), which is constructed in Theorem 6, can be approximated by the solution to the linearized problem (8), (2) as . More precisely, one has the following asymptotic relations: for and , respectively, where is the linear solution and is defined in (56).

5. Asymptotic Linear Profile

In the previous section, we have shown that the solution to the problem (1), (2) can be approximated by the linear solution . The aim of this section is to derive a simpler asymptotic profile of the linear solution .

In Section 2, we derive the solution formula to the problem (8), (9). In the Fourier space, we have , where and are given explicitly in (14) and (15), respectively. First we give the asymptotic expansions of and for . Applying the Taylor expansion to (12), we see that Inserting the previous equalities into (14) and (15), respectively, we arrive at for . Taking Then we find that for , where is a small positive constant.

We now define by Then gives a asymptotic profile of the linear solution . In fact we have the following.

Theorem 8 ( and ). Assume that and , and put . Let be the linear solution, and let be defined by (74). Then one has for .

Proof. Note that , so for the proof of (75), it suffices to show the following estimates: where and , , and are nonnegative integers such that ; we assumed that in the first estimate. These estimates can be proved similarly as in the proof of Lemma 3 by using (73) for and (31), (32), and (72) for . Here we omit the details.

Acknowledgment

This work was supported in part by the NNSF of China (Grant no. 11101144).

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