## Boundary Value Problems

Impact Factor 1.014

Open Access

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

Boundary Value Problems20132013:168

DOI: 10.1186/1687-2770-2013-168

Accepted: 27 June 2013

Published: 16 July 2013

## 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. [25] 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 [46] 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 [714] 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 k th 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 2.1 The solution of the problem (2.2), (2.3) verifies the estimate
(2.10)

for and , 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 2.2 Let and be 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)

for and , 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 2.3 Let and be 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 number such that if and , 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 2.4 Let and . Then we have
(2.23)
(2.24)
(2.25)
and
(2.26)
(2.27)
(2.28)

where in (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 3.1 Assume that is smooth, where is a vector function. Suppose that ( is an integer) when . Then, for the integer , if and , , 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 3.1 Assume that , (). Put
If is suitably small, the Cauchy problem (1.1), (1.2) has a unique global solution satisfying
Moreover, the solution satisfies the decay estimate
(3.4)
and
(3.5)

for and .

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 4.1 Suppose that , (). Put
If is suitably small, the Cauchy problem (1.1), (1.2) has a unique global solution satisfying
Moreover, the solution satisfies the decay estimate
(4.2)
and
(4.3)

for and .

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. □

## Authors’ Affiliations

(1)
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power

## References

1. Cahn JW, Hilliard JE: Free energy of a nonuniform system, I. Interfacial free energy. J. Chem. Phys. 1958, 28: 258–267. 10.1063/1.1744102View Article
2. Galenko P: Phase-field model with relaxation of the diffusion flux in nonequilibrium solidification of a binary system. Phys. Lett. A 2001, 287: 190–197. 10.1016/S0375-9601(01)00489-3View Article
3. Galenko P, Jou D: Diffuse-interface model for rapid phase transformations in nonequilibrium systems. Phys. Rev. E 2005., 71: Article ID 046125
4. Galenko P, Lebedev V: Analysis of the dispersion relation in spinodal decomposition of a binary system. Philos. Mag. Lett. 2007, 87: 821–827. 10.1080/09500830701395127View Article
5. Galenko P, Lebedev V: Local nonequilibrium effect on spinodal decomposition in a binary system. Int. J. Appl. Thermodyn. 2008, 11: 21–28.
6. Gatti S, Grasselli M, Miranville A, Pata V: Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3D. Math. Models Methods Appl. Sci. 2005, 15: 165–198. 10.1142/S0218202505000327
7. Wang Y-X, Wei Z: Global existence and asymptotic behavior of solution to Cahn-Hilliard equation with inertial term. Int. J. Math. 2012., 23: Article ID 1250087
8. Wang Y-X: Existence and asymptotic behavior of solutions to the generalized damped Boussinesq equation. Electron. J. Differ. Equ. 2012, 2012(96):1–11.
9. Wang W, Wu Z: Optimal decay rate of solutions for Cahn-Hilliard equation with inertial term in multi-dimensions. J. Math. Anal. Appl. 2012, 387: 349–358. 10.1016/j.jmaa.2011.09.016
10. Wang Y: Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation. Nonlinear Anal. 2009, 70: 465–482. 10.1016/j.na.2007.12.018
11. Wang Y, Wang Y: Existence and nonexistence of global solutions for a class of nonlinear wave equations of higher order. Nonlinear Anal. 2010, 72: 4500–4507. 10.1016/j.na.2010.02.025
12. Wang Y, Wang Y: Global existence and asymptotic behavior of solutions to a nonlinear wave equation of fourth-order. J. Math. Phys. 2012., 53: Article ID 013512
13. Wang Y, Liu F, Zhang Y: Global existence and asymptotic of solutions for a semi-linear wave equation. J. Math. Anal. Appl. 2012, 385: 836–853. 10.1016/j.jmaa.2011.07.010
14. Wang S, Xu H: On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term. J. Differ. Equ. 2012, 252: 4243–4258. 10.1016/j.jde.2011.12.016View Article
15. Umeda T, Kawashima S, Shizuta Y: On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics. Jpn. J. Appl. Math. 1984, 1: 435–457. 10.1007/BF03167068
16. Li TT, Chen YM: Nonlinear Evolution Equations. Science Press, Beijing; 1989. (in Chinese)
17. Zheng SM: Nonlinear Evolution Equations. CRC Press, Boca Raton; 2004.View Article