Global Existence and Stability of Solution for a p-Kirchhoff type Hyperbolic Equation with Variable Exponents

∆u+ g (ut) = f (u) , (1.2) have been discussed by many authors. For g (ut) = ut, the global existence and blow up results can by found in [14, 18], for g (ut) = |ut| p−2 ut, p > 2, the main results of existence and blow up are in [4, 13]. Many authors studied the existence and nonexistence of solutions for problem with variable exponents, can refer [2, 4, 9, 10, 15, 17, 19, 20]. Messaoudi et al. [13] considered the following equation: utt −∆u+ a |ut| m(x)−2 ut = |u| p(x)−2 u, in Ω× (0, T ) , and used the Faedo Galerkin method to establish the existence of a unique weak local solution. They also proved that the solutions with negative initial energy blow up in finite time. Messaoudi and Talahmeh [11, 12] , considered the following equation:


Introduction
We consider the following boundary value problem: u (x, t) = 0, (x, t) ∈ ∂Ω × (0, T ) , u (x, 0) = u 0 (x) , u t (x, 0) = u 1 (x) x ∈ Ω, (1.1) where Ω is a bounded domain in R n , n ≥ 1 with smooth boundary ∂Ω and M (s) = a + bs with positive parameters a, b, ∆ p u = div(|∇u| p−2 ∇u), with p ≥ 2. r (.) and m (.) are given measurable functions on Ω. Equation (1.1) can be viewed as a generalization of a model introduced by Kirchhoff [6]. The following Kirchhoff type equation have been discussed by many authors. For g (u t ) = u t , the global existence and blow up results can by found in [14,18], for g (u t ) = |u t | p−2 u t , p > 2, the main results of existence and blow up are in [4,13]. Many authors studied the existence and nonexistence of solutions for problem with variable exponents, can refer [2,4,9,10,15,17,19,20]. Messaoudi et al. [13] considered the following equation: u tt − ∆u + a |u t | m(x)−2 u t = |u| p(x)−2 u, in Ω × (0, T ) , and used the Faedo Galerkin method to establish the existence of a unique weak local solution. They also proved that the solutions with negative initial energy blow up in finite time. Messaoudi and Talahmeh [11,12] , considered the following equation: where a, b is a nonnegative constant. They proved a finite-time blow-up result for the solution with negative initial energy as well as for certain solutions with positive initial energy; in the cas where m (x) = 2 and under suitable conditions on the exponents, they established a blow-up result for solutions with arbitrary positive initial energy. Our objective in this paper is to study: In section 2, some notations, assumptions and preliminaries are introduced, section 3 the global existence of solution is proved and the main results of this article are shown in section 4.

Preliminaries
We begin this section with some notations and definitions. Denote by . p , the L p (Ω) norm of a Lebesgue function u ∈ L p (Ω) . We use W 1,p 0 (Ω) to the well-known sobolev space such that u and |∇u| are in L p (Ω) equipped with the norm u W 1,p 0 (Ω) = ∇u p . Let q : Ω → [1, + ∞] be a measurable function, where Ω is adomain of R n . We define the Lebesque space with a variale exponent q (.) by: Next,we define the variable-exponent Sobolev space W 1,q(.) (Ω) as follows: (Ω) such that ∇v exists and |∇v| ∈ L q(.) (Ω) .
For the existence of the local solution of problem (1.1), we refer the reader to [13]. Their result is given in the following theorem: 3 We also assume that m (.) and r (.) satisfy the log-Hölder continuity condition: Then, for any

Exponential growth
In the order to state and prove our result, we define the potential energy functional and the Nehari's functional, respectively, by the following We can considering a = b = 1, and this does not change the general result. and Proof: We multiply the first equation of (1.1) by u t and integrating over the domain Ω, we get Lemma 3.2 Assume that the assumptions of theorem 2.1 and r 1 > 2p, hold, and where Proof: By continuity, there exists T * , such that (3.5) Now, we have for all t ∈ [0, T ] : using (3.5) , we obtain and On the other hand, by Lemma 2.1, we have (Ω) ֒→ L r(.) (Ω) , we obtain This implies that By repeating the above procedure, we can extend T * to T.
So that u t (t)  This implies that the local solution is global in time.

Stability solution
In this section our main result is based a Komornik's inequality [7] , as in [5] . For this, we need the following Lemma: Proof: Multiplying first equation of (1.1) by u (t) E q (t) (q > 0) and integrating over Ω × (S, T ) , we obtain We add and substract the term and use (3.9) , to get Global Existence and Stability of Solution 7 It is clear that Using the definition of E (t) and the following expression Inequality (4.4) , becomes We denote by c the various constants.
We estimate the terms in the right-hand side of (4.5) as follow: Since, 1 ≤ p p−1 < 2, by the embedding of L 2 (Ω) ֒→ L p p−1 (Ω) , we have Thus, by (3.10) , we find For the second term, we have For the third term, we use the following Young inequality: Global Existence and Stability of Solution 9 By (3.3) and Lemma 4.1, we have For the last term of (4.5) , we have This implies First, if we use Young's inequality with λ 1 = (q + 1) /q and λ 2 = q + 1, we have  We take q = m2 2 − 1 to find