- Research Article
- Open Access
Existence and Asymptotic Behavior of Global Solutions for a Class of Nonlinear Higher-Order Wave Equation
© Yaojun Ye 2010
- Received: 5 November 2009
- Accepted: 28 January 2010
- Published: 22 February 2010
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.
- Asymptotic Behavior
- Cauchy Problem
- Global Solution
- Global Existence
- Nonlinear Boundary
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  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.  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  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 , 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 . As in (1.6) is replaced by . Aassila and Guesmia  have obtained a exponential decay theorem through the use of an important lemma of Komornik . Moreover, Messaoudi  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  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  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  proved the existence and uniqueness of classical solutions to the initial boundary problem for (1.1) in Hilbert space. Pecher  investigated the existence and uniqueness of Cauchy problem for (1.1) by the use of the potential well method due to Payne and Sattinger  and Sattinger .
When , for the semilinear higher-order wave equation (1.1), Wang  shows that the scattering operators map a band in into if the nonlinearities have critical or subcritical powers in . Miao  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 .
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 ).
then , where is the maximum time interval on which the solution of problem (1.1)–(1.3) exists.
Please notice that in , we can also construct the following space in proving the existence of local solution by using contraction mapping principle:
which is equipment with norm
Let , and
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.
In order to state and prove our main results, we first define the following functionals:
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.
is a positive constant, where is the most optimal constant in Lemma 2.1, namely, .
Therefore, is a nonincreasing function on .
Suppose that (1.7) holds. If and the initial energy satisfies , then , for each .
Since on , so it holds that
for all .
We obtain from Lemma 2.2 and that
which contradicts (2.15). Thus, we conclude that on .
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).
It follows from Theorem 1.2 that is the global solution of problem (1.1)–(1.3).
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 ).
then , if , and , if , where and are positive constants independent of .
This complete the proof of Lemma 3.2.
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.
for all .
Therefore we conclude from (3.14) and (3.15) that the estimate (3.11) holds.
Therefore we conclude the estimate (3.16) from (3.19).
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.
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).
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