Nonlinear Functional Analysis of Boundary Value Problems 2013View this Special Issue
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Asymptotic Behavior of Global Solutions to the Boussinesq Equation in Multidimensions
The Cauchy problem for the Boussinesq equation in multidimensions is investigated. We prove the asymptotic behavior of the global solutions provided that the initial data are suitably small. Moreover, our global solutions can be approximated by the solutions to the corresponding linear equation as time tends to infinity when the dimension of space .
We investigate the Cauchy problem of the following damped Boussinesq equation in multidimensions: with the initial value Here is the unknown function of and , , and are positive constants. The nonlinear term .
When , (1) has been studied by several authors. The authors investigated the first initial boundary value problem for (1) in a unit circle (see ). The existence and the uniqueness of strong solution were established and the solutions were constructed in the form of series in the small parameter present in the initial conditions. The long-time asymptotic was also obtained in the explicit form. The authors considered the initial-boundary value problem for (1) in the unit ball , similar results were established in .
Recently, Wang  proved the global existence and asymptotic decay of solutions to the problem (1), (2). Their proof is based on the contraction mapping principle and makes use of the sharp decay estimates for the linearized problem. The main purpose of this paper is to establish the following optimal decay estimate of solutions to (1) and (2) by constructing the antiderivatives conditions: Then we obtain a better decay rate of solutions than the previous one in . Moreover, our global solutions can be approximated by the solutions to the corresponding linear equation. The decay estimate is said to be optimal because we have used the sharp decay estimates for the solution operators and , which are defined by (15) and (16), respectively. Since the solution operator has singularity, therefore, we construct the antiderivatives conditions and eliminate the singularity and obtain the same decay estimate for the solution operators . For details; see Lemma 4. The study of the global existence and asymptotic behavior of solutions to wave equations has a long history. We refer to [4–10] for wave equations. Now we state our results as follows.
Theorem 1. Let and let . Assume that . Put If is suitably small, the Cauchy problem (1), (2) has a unique global solution satisfying Moreover, the solution satisfies the decay estimate: for in (6) and in (7).
From the proof of Theorem 1, we have the following corollary immediately.
Corollary 2. Let and assume the same conditions of Theorem 1. Then the solution of the problem (1), (2), which is constructed in Theorem 1, can be approximated by the linear solution as . In fact, we have for and , respectively, where is the linear solution and . Here and are given by (15) and (16), respectively.
Notations. For , denotes the usual Lebesgue space with the norm . The usual Sobolev space of order is defined by with the norm . The corresponding homogeneous Sobolev space of order is defined by with the norm ; when , we write and . We note that for .
The plan of the paper is arranged as follows. In Section 2 we derive the solution formula of the problem (1), (2) and prove the decay property of the solution operators appearing in the solution formula. Then, in Sections 3, we prove the optimal asymptotic decay of solutions to the problem (1), (2).
2. Decay Property
The aim of this section is to derive the solution formula for the problem (1), (2). We first investigate the linear equation of (1): With the initial data (2). Taking the Fourier transform, we have The characteristic equation of (10) is Let be the corresponding eigenvalues of (12), we obtain The solution to the problem (10), (11) is given in the form where We define and by and , respectively, where denotes the inverse Fourier transform. Then, applying to (14), we obtain By the Duhamel principle, we obtain the solution formula to (1), (2) as In what follows, the aim is to establish decay estimates of the solution operators and appearing in the solution formula (18). Firstly, we state the pointwise estimate of solutions in the Fourier space. The result can be found in .
From Lemma 3, we immediately get the following.
Lemma 5. Let be nonnegative integers and let . Then we have for , where in (22). Similarly, we have for .
Proof. We only prove (22). By the Plancherel theorem and (20), 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 (26)–(28) yields (22). We have completed the proof of the Lemma.
Similar to the proof of Lemma 5, it is not difficult to get the following.
Lemma 6. Let and let be nonnegative integers. Then we have the following estimate: for . Similarly, we have for .
3. Proof of Main Result
Lemma 7. Assume that is a smooth function. Suppose that ( is an integer) when . Then for integer , if and , then the following inequalities hold: where .
Proof of Theorem 1. We can prove the existence and uniqueness of small solutions by the contraction mapping principle. Here we only show the decay estimates (6) and (7) for the solution of (18) satisfying with some . Firstly, we introduce the quantity:
We apply the Gagliardo-Nirenberg inequality. This yields
where and . It follows from the definition of in (32) that
provided that . Differentiating (18) times with respect to and taking the norm, we obtain
Firstly, we estimate . We get from (22), with , and for ,
where . By using (23) with , , and to the term , we obtain
Next, we estimate . We divide into two parts and write , where and are corresponding to the time intervals and , respectively. For , making use of (29) with , , and , we arrive at
By Lemma 7, we have the estimates and . Thus by (34), we have
Inserting (39) and (40) into (38) yields
where . Here we assumed . For , exploiting (29) with , , and and using (40), we deduce that
Equations (41) and (42) give
Inserting (36), (37), and (43) into (35), we obtain
for . Consequently, we have , from which we can deduce , provided that is suitably small. This proves the decay estimate (6).
In what follows, we prove (7). Differentiating (18) with respect to and then differentiating the resulting equation times with respect to , we have From (45) and Minkowski inequality, we obtain It follows from (24) that By using (25), we get Finally, we estimate . Dividing into two parts and writing , where and are corresponding to the time intervals and , respectively. Firstly, we estimate the term , applying (30) with , , and and (39), (40), we arrive at Next, for the term , it follows from (30) with , , and and (40) that Collecting (46)–(50), which yields Substituting the estimate into (51), we arrive at the desired estimate (7) for . This completes the proof of Theorem 1.
This work was supported in part by the NNSF of China (Grant no. 11101144) and Innovation Scientists and the Technicians Troop Construction Projects of Henan Province. Funding scheme for young teachers of Universities of Henan Province.
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Copyright © 2013 Yu-Zhu Wang and Qingnian Zhang. 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.