Exponential Growth of Positive Initial Energy Solutions for Coupled Nonlinear Klein-Gordon Equations with Degenerate Damping and Source Terms

The system (1.1) is supplemented with the following initial conditions: ((u (0) , v (0))) = (u0, v0) , ((ut (0) , vt (0))) = (u1, v1) , x ∈ Ω (1.3) and boundary conditions u (x) = v (x) = 0, x ∈ ∂Ω. (1.4) Some special case of the single wave equation with nonlinear damping and nonlinear source terms in the form utt −∆u+ a |ut| p−1 ut = b |u| q−1 u, (1.5) with the presence of different mechanisms of dissipation, damping and for more general forms of nonlinearities has been extensively studied and results concerning existence, nonexistence and asymptotic behavior of solutions have been established by several authors and many results appeared in the literature over the past decades. See ([1] , [5]− [8] , [10] , [16]). The absence of the terms m1u and m 2 2u, equations (1.1) take the form

(1.4) Some special case of the single wave equation with nonlinear damping and nonlinear source terms in the form u tt − ∆u + a |u t | p−1 u t = b |u| q−1 u, (1.5) with the presence of different mechanisms of dissipation, damping and for more general forms of nonlinearities has been extensively studied and results concerning existence, nonexistence and asymptotic behavior of solutions have been established by several authors and many results appeared in the literature over the past decades. See ( [1] , [5] − [8] , [10] , [16]). The absence of the terms m 2 1 u and m 2 2 u, equations (1.1) take the form In [13] Rammaha and Sakuntasathien focus on the global well-posedness of the system of nonlinear wave equation (1.6) . In [17] Wu studied blow up of solutions of the system (1.1) for n = 3 and k = l = θ = ̺ = 0. Agre and Rammaha [3] studied the global existence and the blow up of the solution of problem (1.6) when k = l = θ = ̺, and also Alves et al [4] , investigated the existence, uniform decay rates and blow up of the solution. In [11] Erhen Pişkin prove the blow up of solutions of (1.1) in finite time with negative initial energy and nondegenerate damping terms. In the work [9] , authors considered the following nonlinear viscoelastic system (1.7) and they prove a global nonexistence for certain solutions with positive initial energy, the main tool proof is a method used in [15] . In [14] , B. Said-Houari proved that the energy associated to the system (1.8) (1.8) is unbounded and it grows up as an exponential fonction as time goes to infinity, provided that the initail data are large enough. The key ingredient in his proof is a method used in vitillaro [16] and developed in [15] for a system of wave equations. Our paper is organized as follows, In section 2, we present the assumptions and some lemmas needed for our result. Section 3 is devoted the proof of the main result.

Preliminaries
In this section, we shall give some lemmas which will be used throughout this work.
We assume that We can easily verify that There exist two positive constants c 1 and c 2 such that Proof. By multiplying the first equation of (1.1) by u t and the second equation by v t , integrating over Ω, using integration py parts and summing up, we get Next, we state the local existence theorem that can be established combinig arguments of [12,13] . We give the definition of a weak solution to problem (1.1) − (1.4).

Exponential growth
In this section, we are going to prove our main result. We need in the sequel the following Lemmas.
Proof. Direct computation using Minkowski, Hölder's and Young's inequality and the embedding theorem yields the proof of this Lemma.
We introduce the following constants: where η is the optimal constant in (3.1) .
The following lemma is very useful to prove our result for positive initial energy E (0) > 0. It is similar to the one the lemma in [9], first used by Vitillaro [16] .
By using (2.6) and (3.8) we get Hence, by the above inequality and (2.5) , we have We then define the following Lyaponov function for ǫ small to be chosen later.