- Open Access
Global existence and asymptotic behavior of solutions to a class of fourth-order wave equations
© Zhuang and Zhang; licensee Springer. 2013
Received: 9 May 2013
Accepted: 27 June 2013
Published: 16 July 2013
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.
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 ), 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.
for and . Here is assumed to be small. Compared to the result in , 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.
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 establish decay estimates of the solution operators and appearing in the solution formula (2.8).
for and , where .
which together with (2.14) proves the desired estimates (2.10). Then we have completed the proof of the lemma. □
for and , where .
Substituting the equalities into (2.10) with , we get (2.16).
Substituting the equalities into (2.10) with , we get (2.17). The lemma is proved. □
Equations (2.18) and (2.19) follow from (2.20)-(2.22). The proof of Lemma 2.3 is completed. □
where in (2.23).
where is a positive constant in Lemma 2.3, and and .
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  (see also ).
where , .
where , (i.e., ).
for and .
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)
for and .
Proof Let the mapping Φ be defined in (3.6).
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. □
- 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 ArticleGoogle Scholar
- 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 ArticleGoogle Scholar
- Galenko P, Jou D: Diffuse-interface model for rapid phase transformations in nonequilibrium systems. Phys. Rev. E 2005., 71: Article ID 046125Google Scholar
- 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 ArticleGoogle Scholar
- Galenko P, Lebedev V: Local nonequilibrium effect on spinodal decomposition in a binary system. Int. J. Appl. Thermodyn. 2008, 11: 21–28.Google Scholar
- 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/S0218202505000327MathSciNetView ArticleGoogle Scholar
- 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 1250087Google Scholar
- Wang Y-X: Existence and asymptotic behavior of solutions to the generalized damped Boussinesq equation. Electron. J. Differ. Equ. 2012, 2012(96):1–11.Google Scholar
- 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.2011.09.016MathSciNetView ArticleGoogle Scholar
- Wang Y: Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation. Nonlinear Anal. 2009, 70: 465–482. 10.1016/j.na.2007.12.018MathSciNetView ArticleGoogle Scholar
- 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.2010.02.025MathSciNetView ArticleGoogle Scholar
- 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 013512Google Scholar
- 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.2011.07.010MathSciNetView ArticleGoogle Scholar
- 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 ArticleGoogle Scholar
- 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/BF03167068MathSciNetView ArticleGoogle Scholar
- Li TT, Chen YM: Nonlinear Evolution Equations. Science Press, Beijing; 1989. (in Chinese)Google Scholar
- Zheng SM: Nonlinear Evolution Equations. CRC Press, Boca Raton; 2004.View ArticleGoogle Scholar
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