- Research Article
- Open Access

# Existence and Asymptotic Behavior of Global Solutions for a Class of Nonlinear Higher-Order Wave Equation

- Yaojun Ye
^{1}Email author

**2010**:394859

https://doi.org/10.1155/2010/394859

© Yaojun Ye 2010

**Received:**5 November 2009**Accepted:**28 January 2010**Published:**22 February 2010

## Abstract

The initial boundary value problem for a class of nonlinear higher-order wave equation with damping and source term in a bounded domain is studied, where , is a nature number, and and are real numbers. The existence of global solutions for this problem is proved by constructing the stable sets and shows the asymptotic stability of the global solutions as time goes to infinity by applying the multiplier method.

## Keywords

- Asymptotic Behavior
- Cauchy Problem
- Global Solution
- Global Existence
- Nonlinear Boundary

## 1. Introduction

In this paper we consider the existence and asymptotic behavior of global solutions for the initial boundary problem of the nonlinear higher-order wave equation with nonlinear damping and source term:

where , is a nature number, and are real numbers, is a bounded domain of with smooth boundary , is the Laplace operator, and .

When , the existence and uniqueness, as well as decay estimates, of global solutions and blow up of solutions for the initial boundary value problem and Cauchy problem of (1.1) have been investigated by many people through various approaches and assumptive conditions [1–8]. Rammaha [9] deals with wave equations that feature two competing forces and analyzes the influence of these forces on the long-time behavior of solutions. Barbu et al. [10] study the following initial-boundary value problem:

where is a bounded domain in with a smooth boundary , is a convex, real value function defined on , and denotes the derivative of . They prove that every generalized solution to the above problem and additional regularity blows up in finite time, whenever the exponent is greater than the critical value , and the initial energy is negative.

For the following model of semilinear wave equation with a nonlinear boundary dissipation and nonlinear boundary(interior) sources,

where the operators are Nemytskii operators associated with scalar, continuous functions defined for . The function is assumed monotone. The paper [11, 12] proves the existence and uniqueness of both local and global solutions of this system on the finite energy space and derive uniform decay rates of the energy when .

When , Guesmia [13] considered the equation

with initial boundary value conditions (1.2) and (1.3), where is a continuous and increasing function with , and is a bounded function. He prove a global existence and a regularity result of the problem (1.6), (1.2), and (1.3). Under suitable growth conditions on , he also established decay results for weak and strong solutions. Precisely, In [13], Guesmia showed that the solution decays exponentially if behaves like a linear function, whereas the decay is of a polynomial order otherwise. Results similar to the above system, coupled with a semilinear wave equation, have been established by Guesmia [14]. As in (1.6) is replaced by . Aassila and Guesmia [15] have obtained a exponential decay theorem through the use of an important lemma of Komornik [16]. Moreover, Messaoudi [17] sets up an existence result of this problem and shows that the solution continues to exist globally if ; however, it blows up in finite time if .

Nakao [18] has used Galerkin method to present the existence and uniqueness of the bounded solutions, and periodic and almost periodic solutions to the problem (1.1)–(1.3) as the dissipative term is a linear function . Nakao and Kuwahara [19] studied decay estimates of global solutions to the problem (1.1)–(1.3) by using a difference inequality when the dissipative term is a degenerate case . When there is no dissipative term in (1.1), Brenner and von Wahl [20] proved the existence and uniqueness of classical solutions to the initial boundary problem for (1.1) in Hilbert space. Pecher [21] investigated the existence and uniqueness of Cauchy problem for (1.1) by the use of the potential well method due to Payne and Sattinger [6] and Sattinger [22].

When , for the semilinear higher-order wave equation (1.1), Wang [23] shows that the scattering operators map a band in into if the nonlinearities have critical or subcritical powers in . Miao [24] obtains the scattering theory at low energy using time-space estimates and nonlinear estimates. Meanwhile, he also gives the global existence and uniqueness of solutions under the condition of low energy.

The proof of global existence for problem (1.1)–(1.3) is based on the use of the potential well theory [6, 22]. See also Todorova [7, 25] for more recent work. And we study the asymptotic behavior of global solutions by applying the lemma of Komornik [16].

We adopt the usual notation and convention. Let denote the Sobolev space with the norm , let denote the closure in of . For simplicity of notation, hereafter we denote by the Lebesgue space norm and denotes norm, we write equivalent norm instead of norm . Moreover, denotes various positive constants depending on the known constants and may be different at each appearance.

This paper is organized as follows. In the next section, we will study the existence of global solutions of problem (1.1)–(1.3). Then in Section 3, we are devoted to the proof of decay estimate.

We conclude this introduction by stating a local existence result, which is known as a standard one (see [17]).

Theorem 1.1.

Theorem 1.2.

then , where is the maximum time interval on which the solution of problem (1.1)–(1.3) exists.

Please notice that in [17], we can also construct the following space in proving the existence of local solution by using contraction mapping principle:

which is equipment with norm

We define a distance on , and then is a complete distance space. This show that, for small enough , there exists an unique fixed point on and only depends on . Therefore, with the standard extension method of solution, we obtain for

Here we omit the detailed proof of extension.

## 2. The Global Existence

In order to state and prove our main results, we first define the following functionals:

and according to paper [18, 24] we put

Then, for the problem (1.1)–(1.3), we are able to define the stable set

We denote the total energy related to (1.1) by

for , and is the total energy of the initial data.

Lemma 2.1.

Lemma 2.2.

is a positive constant, where is the most optimal constant in Lemma 2.1, namely, .

Proof.

Lemma 2.3.

Proof.

Therefore, is a nonincreasing function on .

Theorem 2.4.

Suppose that (1.7) holds. If and the initial energy satisfies , then , for each .

Proof.

We obtain from Lemma 2.2 and that

which contradicts (2.15). Thus, we conclude that on .

Theorem.

Assume that (1.7) and (1.8) hold, is a local solution of problem (1.1)–(1.3). If , and , then the solution is a global solution of problem (1.1)–(1.3).

Proof.

It follows from Theorem 1.2 that is the global solution of problem (1.1)–(1.3).

## 3. Decay Estimate

The following two lemmas play an important role in studying the decay estimate of global solutions for the problem (1.1)–(1.3).

Lemma (see [16]).

then , if , and , if , where and are positive constants independent of .

Lemma.

Proof.

This complete the proof of Lemma 3.2.

Theorem.

Let . If one can prove that the energy of the global solution satisfies the estimate

for all , then Theorem 3.3 will be proved by Lemma 3.1. The proof of Theorem 3.3 is composed of the following propositions.

Proposition.

Proof.

Since

Therefore we conclude from (3.14) and (3.15) that the estimate (3.11) holds.

Proposition.

Proof.

Therefore we conclude the estimate (3.16) from (3.19).

Proposition.

Proof.

where and are positive constants depending on and .

and are small enough such that

Therefore, we have from Lemma 3.1 and Proposition 3.6 that

Here is a constant depending on .

It follows from (2.23) and (3.25) that

The proof of Theorem 3.3 is thus finished.

## Declarations

### Acknowledgments

This Research was supported by the Natural Science Foundation of Henan Province (no. 200711013), The Science and Research Project of Zhejiang Province Education Commission (no. Y200803804), The Research Foundation of Zhejiang University of Science and Technology (no. 200803), and the Middle-aged and Young Leader in Zhejiang University of Science and Technology (2008–2012).

## Authors’ Affiliations

## References

- Alves CO, Cavalcanti MM: On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source.
*Calculus of Variations and Partial Differential Equations*2009, 34(3):377–411. 10.1007/s00526-008-0188-zMathSciNetView ArticleMATHGoogle Scholar - Alves CO, Cavalcanti MM, Domingos Cavalcanti VN, Rammaha MA, Toundykov D: On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms.
*Discrete and Continuous Dynamical Systems. Series S*2009, 2(3):583–608.MathSciNetView ArticleMATHGoogle Scholar - Benaissa A, Messaoudi SA: Exponential decay of solutions of a nonlinearly damped wave equation.
*Nonlinear Differential Equations and Applications*2005, 12(4):391–399.MathSciNetView ArticleMATHGoogle Scholar - Georgiev V, Todorova G: Existence of a solution of the wave equation with nonlinear damping and source terms.
*Journal of Differential Equations*1994, 109(2):295–308. 10.1006/jdeq.1994.1051MathSciNetView ArticleMATHGoogle Scholar - Yacheng L, Junsheng Z: On potential wells and applications to semilinear hyperbolic equations and parabolic equations.
*Nonlinear Analysis: Theory, Methods & Applications*2006, 64(12):2665–2687. 10.1016/j.na.2005.09.011MathSciNetView ArticleMATHGoogle Scholar - Payne LE, Sattinger DH: Saddle points and instability of nonlinear hyperbolic equations.
*Israel Journal of Mathematics*1975, 22(3–4):273–303. 10.1007/BF02761595MathSciNetView ArticleMATHGoogle Scholar - Todorova G: Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms.
*Comptes Rendus de l'Academie des Sciences*1998, 326(2):191–196.MathSciNetMATHGoogle Scholar - Vitillaro E: Global nonexistence theorems for a class of evolution equations with dissipation.
*Archive for Rational Mechanics and Analysis*1999, 149(2):155–182. 10.1007/s002050050171MathSciNetView ArticleMATHGoogle Scholar - Rammaha MA: The influence of damping and source terms on solutions of nonlinear wave equations.
*Boletim da Sociedade Paranaense de Matematica*2007, 25(1–2):77–90.MathSciNetMATHGoogle Scholar - Barbu V, Lasiecka I, Rammaha MA: Blow-up of generalized solutions to wave equations with nonlinear degenerate damping and source terms.
*Indiana University Mathematics Journal*2007, 56(3):995–1021. 10.1512/iumj.2007.56.2990MathSciNetView ArticleMATHGoogle Scholar - Cavalcanti MM, Domingos Cavalcanti VN, Lasiecka I: Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping—source interaction.
*Journal of Differential Equations*2007, 236(2):407–459. 10.1016/j.jde.2007.02.004MathSciNetView ArticleMATHGoogle Scholar - Cavalcanti MM, Domingos Cavalcanti VN, Martinez P: Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term.
*Journal of Differential Equations*2004, 203(1):119–158. 10.1016/j.jde.2004.04.011MathSciNetView ArticleMATHGoogle Scholar - Guesmia A: Existence globale et stabilisation interne non linéaire d'un système de Petrovsky.
*Bulletin of the Belgian Mathematical Society*1998, 5(4):583–594.MathSciNetGoogle Scholar - Guesmia A: Energy decay for a damped nonlinear coupled system.
*Journal of Mathematical Analysis and Applications*1999, 239(1):38–48. 10.1006/jmaa.1999.6534MathSciNetView ArticleMATHGoogle Scholar - Aassila M, Guesmia A: Energy decay for a damped nonlinear hyperbolic equation.
*Applied Mathematics Letters*1999, 12(3):49–52. 10.1016/S0893-9659(98)00171-2MathSciNetView ArticleMATHGoogle Scholar - Komornik V:
*Exact Controllability and Stabilization: The Multiplier Method, Research in Applied Mathematics*. Masson, Paris, France; 1994:viii+156.MATHGoogle Scholar - Messaoudi SA: Global existence and nonexistence in a system of Petrovsky.
*Journal of Mathematical Analysis and Applications*2002, 265(2):296–308. 10.1006/jmaa.2001.7697MathSciNetView ArticleMATHGoogle Scholar - Nakao M: Bounded, periodic and almost periodic classical solutions of some nonlinear wave equations with a dissipative term.
*Journal of the Mathematical Society of Japan*1978, 30(3):375–394. 10.2969/jmsj/03030375MathSciNetView ArticleMATHGoogle Scholar - Nakao M, Kuwahara H: Decay estimates for some semilinear wave equations with degenerate dissipative terms.
*Funkcialaj Ekvacioj*1987, 30(1):135–145.MathSciNetMATHGoogle Scholar - Brenner P, von Wahl W: Global classical solutions of nonlinear wave equations.
*Mathematische Zeitschrift*1981, 176(1):87–121. 10.1007/BF01258907MathSciNetView ArticleMATHGoogle Scholar - Pecher H: Die existenz regulärer Lösungen für Cauchy- und anfangs-randwert-probleme nichtlinearer wellengleichungen.
*Mathematische Zeitschrift*1974, 140: 263–279. 10.1007/BF01214167MathSciNetView ArticleMATHGoogle Scholar - Sattinger DH: On global solution of nonlinear hyperbolic equations.
*Archive for Rational Mechanics and Analysis*1968, 30: 148–172.MathSciNetView ArticleMATHGoogle Scholar - Wang B: Nonlinear scattering theory for a class of wave equations in .
*Journal of Mathematical Analysis and Applications*2004, 296(1):74–96. 10.1016/j.jmaa.2004.03.050MathSciNetView ArticleMATHGoogle Scholar - Miao CX: Time-space estimates and scattering at low energy for higher-order wave equations.
*Acta Mathematica Sinica. Series A*1995, 38(5):708–717.MathSciNetGoogle Scholar - Todorova G: Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms.
*Journal of Mathematical Analysis and Applications*1999, 239(2):213–226. 10.1006/jmaa.1999.6528MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.