Growth of Solutions with Positive Initial Energy to Systems of Nonlinear Wave Equations with Damping and Source Terms

where b 1 , b 2 are nonnegative constants and b 1 + b 2 > 0. This type of problems not only is important from the theoretical point of view, but also arises in material science and physics that deal with system of nonlinear wave equations. Ye [1] obtained the local existence and the blowup of the solution of problem (1), for ρ(s) = s. In the absence of the strong damping (Δu t and ΔV t ) terms, problem (1) becomes

Ye [1] obtained the local existence and the blowup of the solution of problem (1), for () =   .In the absence of the strong damping (Δ  and ΔV  ) terms, problem (1) becomes Wu et al. [2] obtained the global existence and blowup of the solution of problem (3) under some suitable conditions.Fei and Hongjun [3] considered problem (3) and improved the blowup result obtained in [2], for a large class of initial data in positive initial energy, using the same techniques as in Payne and Sattinger [4] and some estimates used firstly by Vitillaro [5].Recently, Pis ¸kin and Polat [6,7] studied the local and global existence, energy decay, and blowup of the solution of problem (3).Also, for more information about (1) and (3), see [2,3,7].The many problems associated with (1) are studied from various aspects in many papers [8][9][10][11][12][13].
In this work, we will consider the blowup property in infinity time, that is, exponential growth.This work is organized as follows.In Section 2, we state the local existence result.In Section 3, we establish that the energy will grow up as an exponential as time goes to infinity, provided that the initial data are large enough or (0) <  1 , where (0) and  1 are defined in ( 9) and (15).

Preliminaries
In this section, we introduce some notations and lemmas and local existence theorem needed in the proof of our main results.Let ‖ ⋅ ‖ and ‖ ⋅ ‖  denote the usual  2 (Ω) norm and   (Ω) norm, respectively.
Concerning the functions  1 (, V) and  2 (, V), we take where ,  > 0 are constants and  satisfies According to the above equalities they can easily verify that where We have the following result.

Exponential Growth
In this section, we will prove that the energy is unbounded when the initial data are large enough in some sense.Our techniques of proof follow very carefully the techniques used in [16].
For the sake of simplicity and to prove our result, we take  =  = 1 and introduce where  is the optimal constant in (14).Next, we will state a lemma which is similar to the one introduced firstly by Vitillaro in [5] to study a class of a single wave equation.
We consider the following functional: for small  to be specified later.
Our goal is to show that () satisfies a differential inequality of the form This, of course, will lead to exponential growth.By taking a derivative of (22) and using (1), it follows that From ( 9) and (20), it follows that −  2 (‖∇‖ Then using (18), we obtain where . In order to estimate the last two terms in (27), we use the following Young inequality: where ,  ≥ 0,  > 0, ,  ∈  + such that 1/ + 1/ = 1.
Remark 8.When (0) < 0, by setting () = −(), the similar result is obtained by applying the same arguments in the proof of Theorem 7.