On Behaviors of the Energy of Solutions for Some Damped Nonlinear Hyperbolic Equations with p-Laplacian

where∆pu = div(|∇xu|∇xu) and p ≥ 2 is real number,Ω is a bounded domain inR with smooth boundary ∂Ω and the real numbers ω,m and r satisfy appropriate conditions to be made precise in the sequel. Several authors have studied the global existence and asymptotic behavior of solutions related to the problem (P ) (see for instance [2], [4], [5], [10], [19] and [20]. In all this above cited papers the damping term played an important role in order to give energy decay estimates. In the case where σ ≡ 1 with considering (−∆)ut instead of damping term σ(t)(ut − ∆ut) in the probleme (P ), Gao and Ma [5] obtained global existence results by means of the Faedo-Galerkin approximations. Further they shown the asymptotic behavior of solutions through the use the integral inequality given by Nakao [11]. However, it will be difficult to proceed by this method with more general functions σ. Also, Chen, Yao and Shao [4] investigated the global existence and uniqueness of a solution to an initial boundary problem utt − ∆pu − ∆ut + g(x, u) = f(x). There they established a polynomial decay of energy under certain assumptions on g where 2 ≤ p < n. See also Ye [14, 15], Ma and Soriano [7] for related results. It is worthmentioning some other papers in connectionwith asymptotic behavior of solutions to the nonlinear hyperbolic equation with dissipative effects, e.g., [1], [3], [9], [13], [16] and the references therein. Inspired by [4], we investigated in this paper the decay rate estimate for the energy of the global solutions to the problem (P ). For our purpose, we use themultiplier method combinedwith a nonlinear integral inequalities given by Martinez [8] which depends on the construction of a special weight function that depends on the behavior of σ. The paper is organized as follows. In the next section, we present some notations and material needed for our work. The statement and the proof of our main result will be given in section 3. To simplify notation, we often write u(t) instead u(x, t) and ut(t) instead ut(x, t). The norm in Lebesgue space L(Ω) is denoted by ∥ ·∥p, in particular ∥ ·∥2 denotes L(Ω). We also write equivalent norm ∥∇.∥p instead ofW 1,p 0 (Ω) norm ∥ · ∥W 1,p 0 (Ω) and throughout this paper the functions considered are all real valued. International Journal of Advanced Research in Mathematics Submitted: 2016-08-24 ISSN: 2297-6213, Vol. 6, pp 13-20 Revised: 2016-09-07 doi:10.18052/www.scipress.com/IJARM.6.13 Accepted: 2016-09-08 2016 SciPress Ltd., Switzerland Online: 2016-09-30


Introduction
This paper deals with the decay rate estimate for the energy of the problem where ∆ p u = div(|∇ x u| p−2 ∇ x u) and p ≥ 2 is real number, Ω is a bounded domain in R n with smooth boundary ∂Ω and the real numbers ω, m and r satisfy appropriate conditions to be made precise in the sequel.
Several authors have studied the global existence and asymptotic behavior of solutions related to the problem (P ) (see for instance [2], [4], [5], [10], [19] and [20]. In all this above cited papers the damping term played an important role in order to give energy decay estimates. In the case where σ ≡ 1 with considering (−∆ α )u t instead of damping term σ(t)(u t − ∆u t ) in the probleme (P ), Gao and Ma [5] obtained global existence results by means of the Faedo-Galerkin approximations. Further they shown the asymptotic behavior of solutions through the use the integral inequality given by Nakao [11]. However, it will be difficult to proceed by this method with more general functions σ.
Also, Chen, Yao and Shao [4] investigated the global existence and uniqueness of a solution to an initial boundary problem u tt − ∆ p u − ∆u t + g(x, u) = f (x). There they established a polynomial decay of energy under certain assumptions on g where 2 ≤ p < n. See also Ye [14,15], Ma and Soriano [7] for related results.
Inspired by [4], we investigated in this paper the decay rate estimate for the energy of the global solutions to the problem (P ). For our purpose, we use the multiplier method combined with a nonlinear integral inequalities given by Martinez [8] which depends on the construction of a special weight function that depends on the behavior of σ.
The paper is organized as follows. In the next section, we present some notations and material needed for our work. The statement and the proof of our main result will be given in section 3.
To simplify notation, we often write u(t) instead u(x, t) and u t (t) instead u t (x, t). The norm in Lebesgue space L p (Ω) is denoted by ∥ · ∥ p , in particular ∥ · ∥ 2 denotes L 2 (Ω). We also write equivalent norm ∥∇.∥ p instead of W 1,p 0 (Ω) norm ∥ · ∥ W 1,p 0 (Ω) and throughout this paper the functions considered are all real valued.

Preliminaries
First, suppose that σ : R + → R + is a non increasing positive function of class C 1 on R + , satisfying (1) We denote the total energy functional associated to the solutions of the problem (P ) by Before stating our main result, we briefly recall the following result on the existence of a solution of the problem (P ).
This result can be established by using Faedo-Galerkin method. The proof closely follows the argument presented in [4], [6] and [13].
We now present some useful lemmas which will be used later.

Lemma 2. Let u(x, t) be a global solution to the problem (P ) on [0, ∞). Then we have,
This Lemma can be easily proved by multiplying the both sides of the first equation of (P ) by u t , integrating over Ω and then using integration by parts.
In order to solve the energy decay of the problem (P ), we use the following lemma.
Lemma 4 [8]. Let E : R + → R + be a non increasing function and ϕ : Assume that there exist q ≥ 0 and γ > 0 such that Then we have

Main results and proof
We are now ready to state and prove our main result.
Theorem 5. Let (u 0 , u 1 ) ∈ W 1,p 0 (Ω) × L 2 (Ω) and n > p > 2. Suppose that (1) holds. Assume further that p ≤ r ≤ 2p and r < m < np n−p . Then there exists a positive constant c(E(0)) depending continuously on E(0) such that the solution u(x, t) of the problem (P ) satisfies the following energy decay estimate Proof. Multiplying by E q ϕ ′ (t) u on both sides of the first equation of (P ) and integrating over Ω × [T, S], we obtain that where 0 ≤ S ≤ T ≤ +∞ and ϕ is a function satisfying all the hypotheses of Lemma 4.
By an integration by parts we see that Hence from the definition of energy and a simple computation we get We must estimate every terms of right-hand side of (6) to arrive at a similar inequality as (4). Define,

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so that ϕ is a nondecreasing function of class C 2 on R + and the hypothesis (1) ensures that Exploiting Cauchy-Schwartz inequality, Sobolev-poincaré inequality and the definition of energy we get ∫ Using nonincreasing property of E and the fact that ϕ ′ is a bounded non negative function on R + (we denote by µ its maximum) we obtain that here and from now on, c denotes a positive constant which can be different from line to line.
Similarly we have On the other hand, from Lemma 2 we have that We also need to estimate From Sobolev-Poincaré inequality, there exists r ′ > 0 such that We also notice that there exists ω ′ > 0 where ω > ω ′ so that,

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Thus, we obtain from (11) that and taking into account (12), we have Using the definition of the energy E(t) we see that ∫ Consequently, The remaining term of the right hand side of (6) can be estimate as follows, We received from Hölder inequality and Sobolev-Poincaré inequality that We also have This gives Further, by Young inequality, we have for ε > 0 Thus takes, 2 = q + 1, so that q = (p − 2)/p. Then substituting the estimates (8), (9), (10), (14) and (13) into (6), we get where c, c ′ and c ′′ are different positive constants independent of E(0). Let T → +∞, we have from (15) that Thus we receive from Lemma 4 that is a positive constant depending on E(0). As q = (p − 2)/p, we have E(t) ≤ (c(E(0))) The proof is thus finished.