Existence and Decay Estimate of Global Solutions to Systems of Nonlinear Wave Equations with Damping and Source Terms

The initial-boundary value problem for a class of nonlinear wave equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set and obtain the asymptotic stability of global solutions through the use of a difference inequality.


Introduction
In this paper, we are concerned with the global solvability and decay stabilization for the following nonlinear wave equations system: with the initial-boundary value conditions ( , ) = 0, V ( , ) = 0, ( , ) ∈ Ω × + , where Ω is a bounded open domain in with a smooth boundary Ω, , ≥ 2, > 0 and < 2( +2) ≤ /( − ) for ≥ and < 2( + 2) < +∞ for < . When = 2, Medeiros and Miranda [1] proved the existence and uniqueness of global weak solutions. Cavalcanti et al. in [2][3][4] considered the asymptotic behavior for wave equation and an analogous hyperbolic-parabolic system with boundary damping and boundary source term. In paper [5,6], the authors dealt with the existence, uniform decay rates, and blowup for solutions of systems of nonlinear wave equations with damping and source terms.
Rammaha and Wilstein [7] and Yang [8] are concerned with the initial boundary value problem for a class of quasilinear evolution equations with nonlinear damping and source terms. Under appropriate conditions, by a Galerkin approximation scheme combined with the potential well method, they proved the existence and asymptotic behavior of global weak solutions when < , where ≥ 0 and are, respectively, the growth orders of the nonlinear strain terms and the source term.
For the following strongly damped nonlinear wave equation Dell'Oro and Pata [10] obtain the long-time behavior of the related solution semigroup, which is shown to possess the global attractor in the natural weak energy space. In addition, the existence of global and local solutions, decay estimates, and blowup for solutions of nonlinear wave equation with source and damping terms and exponential nonlinearities are studied in [11][12][13][14].
In this paper, we prove the global existence for the problem (1)-(5) by applying the potential well theory introduced by Sattinger [15] and Payne and Sattinger [16]. Meanwhile, we obtain the asymptotic stabilization of global solutions by using a difference inequality [17].
Definition 3. A pair of functions ( , V) is said to be a weak solution of (1)- for all test functions , ∈ where Proof. Let { } ∞ =1 be a basis for 1, 0 (Ω). Supposed that is the subspace of 1, 0 (Ω) generated by { 1 , 2 , . . . , }, ∈ . We are going to look for the approximate solution Abstract and Applied Analysis 3 which satisfies the following Cauchy problem: ∫ Note that, we can solve the problem (14)-(19) by a Picard's iteration method in ordinary differential equations. Hence, there exists a solution in [0, ) for some > 0, and we can extend this solution to the whole interval [0, ] for any given > 0 by making use of the a priori estimates below. Multiplying (14) by ( ) and (15) by ℎ ( ) and summing over from 1 to , we obtain By summing (20) and (21) and integrating the resulting identity over [0, ], we have We estimate the right-hand terms of (22) as follows: we get from Hölder inequality and Lemmas 1 and 2 that It follows from (22) and (23) that which implies that We get from (25) and Gronwall type inequality that Abstract and Applied Analysis Thus, we deduce from (26) that there exists a time > 0 such that where 1 is a positive constant independent of . We have from (24) and (26) that It follows from (27) and (28) that and 2 ([0, ] ; 1 0 (Ω)) . (29) Using the same process as the proof of Theorem 2.1 in paper [18], we derive that [ ( ), V( )] is a local solution of the problem (1)- (5). By (20) and (21), we conclude that (11) is valid.

Global Existence
In order to state our main results, we first introduce the following functionals: Then, we are able to define the stable set as follows for problem (1)-(5): We denote the total energy related to (1) and (2) by (12), and is the total energy of the initial data.
We have from (11) that ( ) is the primitive of an integrable function. Therefore, ( ) is absolutely continuous, and equality (35) is satisfied.
It follows from Hölder inequality and Lemma 1 that We get from (39) We conclude from (41) and (43) that Thus, we complete the proof of Lemma 6.
Proof. It suffices to show that ‖ ‖ 2 + ‖V ‖ 2 + ‖∇ ‖ + ‖∇V‖ is bounded uniformly with respect to . Under the hypotheses in Theorem 8, we get from Lemma 7 that [ , V] ∈ on [0, ). So the following formula holds on [0, ): We have from (51) that Hence, we get The above inequality and the continuation principle lead to the global existence of the solution [ , V] for problem (1)-(5).

Asymptotic Behavior of Global Solutions
The following lemma plays an important role in studying the decay estimate of global solutions for the problem (1)-(5).