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# Global existence and asymptotic behavior of solutions to a class of fourth-order wave equations

Zhitao Zhuang and Yuanzhang Zhang*

Author Affiliations

School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China

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Boundary Value Problems 2013, 2013:168  doi:10.1186/1687-2770-2013-168

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/168

 Received: 9 May 2013 Accepted: 27 June 2013 Published: 16 July 2013

© 2013 Zhuang and Zhang; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This paper is concerned with the Cauchy problem for a class of fourth-order wave equations in an n-dimensional space. Based on the decay estimate of solutions to the corresponding linear equation, a solution space is defined. We prove the global existence and optimal decay estimate of the solution in the corresponding Sobolev spaces by the contraction mapping principle provided that the initial value is suitably small.

MSC: 35L30, 35L75.

##### Keywords:
fourth-order wave equation; global existence; decay estimate

### 1 Introduction

We investigate the Cauchy problem for a class of fourth-order wave equations

(1.1)

with the initial value

(1.2)

Here is the unknown function of and , and is a constant. The nonlinear term is a smooth function with for .

Equation (1.1) is reduced to the classical Cahn-Hilliard equation if (see [1]), which has been widely studied by many authors. Galenko et al.[2-5] proposed to add inertial term to the classical Cahn-Hilliard equation in order to model non-equilibrium decompositions caused by deep supercooling in certain glasses. For more background, we refer to [4-6] and references therein. It is obvious that (1.1) is a fourth-order wave equation. For global existence and asymptotic behavior of solutions to more higher order wave equations, we refer to [7-14] and references therein.

Very recently, global existence and asymptotic behavior of solutions to the problem (1.1), (1.2) were established by Wang and Wei [7] under smallness condition on the initial data. When , , they obtained the following decay estimate:

(1.3)

for and . The main purpose of this paper is to refine the result in [7] and prove the following decay estimate for the solution to the problem (1.1), (1.2) for with data,

(1.4)

for and . Here is assumed to be small. We also establish the decay estimate for the solution to the problem (1.1), (1.2) for with data,

(1.5)

for and . Here is assumed to be small. Compared to the result in [7], we obtain a better decay estimate of solutions for small initial data.

The paper is organized as follows. In Section 2, we study the decay property of the solution operators appearing in the solution formula. We prove global existence and asymptotic behavior of solutions for the Cauchy problem (1.1), (1.2) in Sections 3 and 4, respectively.

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

We denote its inverse transform by . For , denotes the usual Lebesgue space with the norm . The usual Sobolev space of order s is defined by with the norm . The corresponding homogeneous Sobolev space of order s is defined by with the norm ; when , we write and . We note that for .

For a nonnegative integer k, denotes the totality of all the kth order derivatives with respect to . Also, for an interval I and a Banach space X, denotes the space of k-times continuously differential functions on I with values in X.

Throughout the paper, we denote every positive constant by the same symbol C or c without confusion. is the Gauss symbol.

### 2 Decay property

The aim of this section is to derive the solution formula to the Cauchy problem (1.1), (1.2). Without loss of generality, we take . We first study the linearized equation of (1.1),

(2.1)

with the initial data in (1.2). Taking the Fourier transform, we have

(2.2)

The corresponding initial value are given as

(2.3)

The characteristic equation of (2.2) is

Let be the corresponding eigenvalues, we obtain

(2.4)

The solution to the problem (2.2), (2.3) is given in the form

(2.5)

where

(2.6)

and

(2.7)

We define and by and , respectively, where denotes the inverse Fourier transform. Then, applying to (2.5), we obtain

(2.8)

By the Duhamel principle, we obtain the solution formula to (1.1), (1.2)

(2.9)

The aim of this section is to establish decay estimates of the solution operators and appearing in the solution formula (2.8).

Lemma 2.1The solution of the problem (2.2), (2.3) verifies the estimate

(2.10)

forand, where.

Proof We apply the energy method in the Fourier space to prove (2.10). Such an energy method was first developed in [15]. We multiply (2.2) by and take the real part. This yields

(2.11)

Multiplying (2.2) by and taking the real part, we obtain

(2.12)

Combining (2.11) and (2.12) yields

(2.13)

where

and

A simple computation implies that

(2.14)

where

Note that

It follows from (2.14) that

(2.15)

where

Using (2.13) and (2.15), we get

Thus

which together with (2.14) proves the desired estimates (2.10). Then we have completed the proof of the lemma. □

Lemma 2.2Letandbe the fundamental solutions to (2.1) in the Fourier space, which are given in (2.6) and (2.7), respectively. Then we have the estimates

(2.16)

and

(2.17)

forand, where.

Proof If , from (2.5), we obtain

Substituting the equalities into (2.10) with , we get (2.16).

In what follows, we consider , it follows from (2.5) that

Substituting the equalities into (2.10) with , we get (2.17). The lemma is proved. □

Lemma 2.3Letandbe the fundamental solutions to (2.1) in the Fourier space, which are given in (2.6) and (2.7), respectively. Then there exists a small positive numbersuch that ifand, we have the following estimate:

(2.18)

and

(2.19)

Proof For sufficiently small ξ, using the Taylor formula, we get

(2.20)

and

(2.21)

It follows from (2.6) and (2.7) that

(2.22)

Equations (2.18) and (2.19) follow from (2.20)-(2.22). The proof of Lemma 2.3 is completed. □

Lemma 2.4Letand. Then we have

(2.23)

(2.24)

(2.25)

and

(2.26)

(2.27)

(2.28)

wherein (2.23).

Proof By the property of the Fourier transform and (2.16), we obtain

(2.29)

where is a positive constant in Lemma 2.3, and and .

By a straight computation, we get

(2.30)

It follows from the Hausdorff-Young inequality that

(2.31)

Combining (2.29)-(2.31) yields (2.23). Similarly, we can prove (2.24)-(2.28). Thus we have completed the proof of the lemma. □

### 3 Global existence and decay estimate (I)

The purpose of this section is to prove global existence and asymptotic behavior of solutions to the Cauchy problem (1.1), (1.2) with data. We need the following lemma, which comes from [16] (see also [17]).

Lemma 3.1Assume thatis smooth, whereis a vector function. Suppose that (is an integer) when. Then, for the integer, ifand, , then. Furthermore, the following inequalities hold:

(3.1)

and

(3.2)

where, .

Based on the decay estimates of solutions to the linear problem (2.1), (1.2), we define the following solution space:

where

For , we define

The Gagliardo-Nirenberg inequality gives

(3.3)

where , (i.e., ).

Theorem 3.1Assume that, (). Put

Ifis suitably small, the Cauchy problem (1.1), (1.2) has a unique global solutionsatisfying

Moreover, the solution satisfies the decay estimate

(3.4)

and

(3.5)

forand.

Proof Let us define the mapping

(3.6)

Using (2.23), (2.24), (2.27), (3.1) and (3.3), for , we obtain

Thus

(3.7)

It follows from (3.6) that

(3.8)

By exploiting (3.8), (2.25), (2.26), (2.28), (3.1) and (3.3), for , we have

The above inequality implies

(3.9)

Combining (3.7) and (3.9) and taking and R suitably small, we get

(3.10)

For , (3.6) gives

(3.11)

By (2.27), (3.2) and (3.3), for , we infer that

which implies

(3.12)

Similarly, for , from (3.11), (2.28) and (3.2), (3.3), we deduce that

which gives

(3.13)

Combining (3.12) and (3.13) and taking R suitably small yields

(3.14)

From (3.10) and (3.14), we know that Φ is a strictly contracting mapping. Consequently, we conclude that there exists a fixed point of the mapping Φ, which is a solution to (1.1), (1.2). We have completed the proof of the theorem. □

### 4 Global existence and decay estimate (II)

In the previous section, we have proved global existence and asymptotic behavior of solutions to the Cauchy problem (1.1), (1.2) with data. The purpose of this section is to establish global existence and asymptotic behavior of solutions to the Cauchy problem (1.1), (1.2) with data. Based on the decay estimates of solutions to the linear problem (2.1), (1.2), we define the following solution space:

where

For , we define

Thanks to the Gagliardo-Nirenberg inequality, we get

(4.1)

Theorem 4.1Suppose that, (). Put

Ifis suitably small, the Cauchy problem (1.1), (1.2) has a unique global solutionsatisfying

Moreover, the solution satisfies the decay estimate

(4.2)

and

(4.3)

forand.

Proof Let the mapping Φ be defined in (3.6).

For , (2.23), (2.24), (2.27), (3.1) and (4.1) give

Thus we get

(4.4)

Applying to (3.6), we obtain

(4.5)

By using (2.25), (2.26), (2.28), (3.1), (4.1), for , we have

This yields

(4.6)

Combining (4.4) and (4.6) and taking and R suitably small, we obtain

(4.7)

For , by using (3.6), we have

(4.8)

It follows from (2.27), (3.2) and (4.1) for that

which implies

(4.9)

Similarly, for , from (4.5), (2.28), (3.2) and (4.1), we infer that

which implies

(4.10)

Using (4.9) and (4.10) and taking R suitably small yields

(4.11)

It follows from (4.7) and (4.11) that Φ is a strictly contracting mapping. Consequently, we infer that there exists a fixed point of the mapping Φ, which is a solution to (1.1), (1.2). We have completed the proof of the theorem. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed to each part of this work equally and read and approved the final manuscript.

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