EXISTENCE AND DECAY OF SOLUTIONS FOR A HIGHER-ORDER VISCOELASTIC WAVE EQUATION WITH LOGARITHMIC NONLINEARITY

u (x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ Ω, ∂ ∂νiu (x, t) = 0, (i = 1, 2, ...,m− 1) x ∈ ∂Ω× (0, T ) , (1) where Ω ⊂ R is a bounded domain with smooth boundary ∂Ω, v is the unit outer normal, k is positive constant to be chosen later and P = (−4) , (m ≥ 1 and m ∈ N). The kernel g has some conditions to be specified later. The equation with the logarithmic source term is related with many branches of physics. Cause of this is interest in it occures naturally in inflation cosmology and supersymmetric field theories, quantum mechanics, nuclear physics [6, 8, 9]. Some of them authors [3—5,11,15,16] improve many results in the literature.

The equation with the logarithmic source term is related with many branches of physics. Cause of this is interest in it occures naturally in in ‡ation cosmology and supersymmetric …eld theories, quantum mechanics, nuclear physics [6,8,9]. Some of them authors [3-5, 11, 15, 16] improve many results in the literature.

EXISTENCE AND DECAY OF SO LUTIONS FOR A VISCOELASTIC WAVE EQUATION 301
When m = 2; problem (1) becomes the following ju t j u tt + 2 u tt + 2 u t Z 0 g (t s) 2 u ds + u = u ln juj k : In [2], Al-Gharabli et al. investigated the local existence, global existence and stability for the problem (2). In [13], Peyravi consider in R 3 with h (s) = k 0 + k 1 jsj m 1 : He studied the decay estimate and exponential growth of solutions for the problem (3).
Motivated by the above studies, we asked that what results will be obtained if one revises the Laplace operator by other high order viscoelastic term. Then, we established the local existence, and general decay estimates of the solution for problem (1).
The rest of our work is organized as follows. In section 2, we give some notations and lemmas which will be used throughout this paper. In section 3, our purpose is to get suitable conditions of the local existence the solutions of the problem. In section 4, we established the general decay of the solutions of the problem.

Preliminaries
In this part, we give some notations and lemmas and preliminary results in order to state the main results of this paper. We use the standart Lebesgue space L p ( ) and Sobolev space H m ( ) with their scalar products and norms. Meanwhile we de…ne H m 0 ( ) = n u 2 H m ( ) : @ i u @v i = 0; i = 0; 1; :::m 1 o and introduce the following abbreviations; k:k = k:k L 2 ( ) , k:k p = k:k L p ( ) and k:k H m = k:k H m ( ) (For detailed information about these spaces, see [1,14]). We denote by C and C i (i = 1; 2; :::) various positive constants. Now we give some important lemmas for proof of our theorems.
where ' 1 and b > 0 are positive constants. Then we have Now, we present following assumptions: (A3) There exist a nonincreasing di¤erentiable function ! : R + ! R + such that (A4) The constant k in (1) satis…es 0 < k < k 1 ;where k 1 is the positive real number satisfying and c p is de…ned in Corollary 2.
Lemma 5. The energy functional E (t) is decreasing with respect to t. Where

EXISTENCE AND DECAY OF SO LUTIONS FOR A VISCOELASTIC WAVE EQUATION 303
Proof. We multiply both sides of (1) by u t and then integrating over Next, we begin with de…ning the potential energy functional and Nehari functional on H m 0 ( ) 8 > > > > < > > > > :

Local existence
In this section we state and prove the local existence result for problem (1). The proof is based on Faedo-Galerkin method. Proof. We will use the Faedo-Galerkin method to construct approximate solutions. Let fw j g 1 j=1 be an orthogonal basis of the "separable" space H m 0 ( ). Let V m = span fw 1 ; w 2 ; :::; w m g ; and let the projections of the initial data on the …nite dimensional subspace V m be given by for j = 1; 2; :::; m:

EXISTENCE AND DECAY OF SO LUTIONS FOR A VISCOELASTIC WAVE EQUATION 305
We look for the approximate solution This leads to a system of ordinary di¤erantial equations for unknown functions h m j (t). Based on standard existence theory for ordinary di¤erantial equation, one can obtain functions h j : [0; t m ) ! R; j = 1; 2; :::; m; which satisfy (14) in a maximal interval [0; t m ) ; 0 < t m T: Next, we show that t m = T and that the local solution is uniformly bounded independent of m and t. For this purpose, let us replace w by u m t in (14) d dt so that we can write Then by integrating (16) with respect to t from 0, we obtain by combining (17) and (18), we obtain where C = 2E m (0): Choosing e We know that We make use of the following Cauchy-Schwarz inequality

EXISTENCE AND DECAY OF SO LUTIONS FOR A VISCOELASTIC WAVE EQUATION 307
we obtain Then by (20) we have where Noting that x ln x (x + B) ln (x + B) for any x > 0; B 1 holds, then by using of Logarithmic Gronwall inequality, we obtain Hence, from inequality (23) and (20) where C 2 is a positive constant independent of m and t. So, the approximate solution is uniformly bounded independent of m and t. Therefore, we can extend t m to T .
Substituting w = u m tt in (14) and using of Z uu tt dx ku tt k 2 + 1 4 kuk 2 inequality, we get To estimate the last term of (25), we will use (6) with 0 = 1 2 and Young's, Cauchy-Schwarz and the Embedding inequalities, we obtain

EXISTENCE AND DECAY OF SO LUTIONS FOR A VISCOELASTIC WAVE EQUATION 309
From the last inequality, if we take = min f 2 ; c 4 ; 3 ; 1 g > 0 small enough and using (24), we have the following, for some C 3 > 0 not depending m or t : From (24) and (28), we obtain 8 < : u m ; is uniformly bounded in L 1 (0; T ; H m 0 ( )) ; u m t ; is uniformly bounded in L 1 (0; T ; H m 0 ( )) ; u m tt ; is uniformly bounded in L 2 (0; T ; H m 0 ( )) ; that there exists a subsequence of (u m ) (still denoted by (u m )), such that 8 > > > > < > > > > : Then using (29) and Aubin-Lions'lemma, we have Since the map s ! s ln jsj k is continuous, we have the convergence By the Sobolev embedding theorem (H 2 0 ( ) ,! L 1 ( )), it is clear that u m ln ju m j k u ln juj k is bounded in L 1 ( (0; T )) : Next, taking into account the Lebesgue bounded convergence theorem, we have u m ln ju m j k ! u ln juj k ; strongly in L 2 0; T ; L 2 ( ) : We integrate (14) over (0; t) to obtain, 8w 2 V m Convergences (13), (30) ,(32) are su¢ cient to pass to the limit in (33) as m ! 1; which implies that (34) is valid 8w 2 H m 0 ( ) :Using the fact that the terms in the right-hand side of (34) are absolutely continuous since they are functions of t de…ned by integrals over (0; t), hence it is di¤erentiable for a.e. t 2 R + . Thus, di¤erentiating (34), we obtain, for a.e. t 2 (0; T ) and any w 2 H m 0 ( ), Z u tt wds + This completed the proof.

General Decay
In this section we study general decay of problem (1). Now, we introduce the following For the logarithmic source, we assume that k 1: The next lemma by Martinez plays an important role in our proof. Lemma 8. [12] Let E : R + ! R + be a nonincresing function and : R + ! R + be a C 2 increasing function such that (0) = 0 and lim t!1 (t) = 1:Assume that there exists c > 0 for which and then E (t) E (0) e w (t) ; for some positive constants w and : Theorem 9. Suppose that (A1) (A3) hold. Let ku 0 (x)k 2 < and 0 < E (0) < E 1 : Then, there exist two positive constants n and ñ suh that (36) holds for all t > 0: To prove the our theorem we need the following lemma.

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Similary, we get Also, we have For the …rst term in right side of (48), we use (42) and Young inequality to obtain For the estimating last term of (44) using Young's inequality, (42) and (7), for " > 0; we obtain Z P