Decay Rate for a Viscoelastic Equation with Strong Damping and Acoustic Boundary Conditions

This paper is concerned with a nonlinear viscoelastic equation with strong damping: ( ) ( ) 0 , d 0, t t tt tt t u u u u g t s u x s s u ρ − ∆ − ∆ + − ∆ − ∆ = ∫ . The objective of the present paper is to provide some results on the long-time behavior to this equation with acoustic boundary conditions. By using the assumptions on the relaxation function due to Tatar [1], we show an arbitrary rate of decay with not necessary of an exponential or polynomial one and without the assumption ( ) 0 1 d 2 g s s ∞ < ∫ condition. The result extends and improves some results given in Cavalcanti [2].

 , ν is the unit outward normal to Γ , the function g represents the kernel of a memory, p and q are specific functions, and ρ is a real number such that Our problem is of the form ( ) which has several modeling features.In the case, ( ) t f u is a constant; Equation (8) has been used to model extensional vibrations of thin rods (see Love [3], Chapter 20).In the case, ( ) t f u is not a constant; Equation ( 8) can model ma- terials whose density depends on the velocity t u , for instance, a thin rod which possesses a rigid surface and with an interior which can deform slightly.We refer the reader to Fabrizio and Morro [4] for several other related models.
Recently, Liu [5] considered the following viscoelastic problem with acoustic boundary conditions ,0 , ,0 , , the authors obtain an arbitrary decay rate of the energy.In the pioneering paper [6], Beale and Rosencrans considered the acoustic boundary condition (1.12) and the coupled impenetrability boundary condition (1.11) with a general form, which had the presence of tt y in (1.2), in a study of the model for acoustic wave motion of a fluid interacting with a so-called locally reacting surface.Recently, many authors treated wave equations with acoustic boundary conditions, see [7] [8] [9] [10] and references therein.For instance, Rivera and Qin [10] proved the polynomial decay for the wave motion with general acoustic boundary conditions by using the Lyapunov functional technique.Frota and Larkin [8] established global solvability and the exponential decay for problems (1.9)-(1.13)with 0 g ≡ .They overcame the difficulties which were arisen due to the absence of tt y in (1.12) by using the degenerated second order equation.Recently, Park and Park [9] investigated problems (1.9)-(1.13)and proved general rates of decay which depended on the behavior of g , under the additional assumption of that ( ) 0 d .g s s +∞ ∫ coelastic problem were proved but no rate of decay has been specified.Since then problems related to viscoelasticity have attracted a great deal of attention [13]  In the case Messaoudi and Tatar [17] proved the exponential decay of global solutions to (15) without smallness of initial data, considering only the dissipation effect given by the memory.
In [18] [19], the condition has been replaced by ( ) ( ) ( ) with the Dirichlet boundary condition, where 0 ρ > is a constant.We also mention that Fabrizio and Polidoro [22] obtained the exponential decay result under the conditions that Recently, Tatar [23] improved these results by removing the last condition and established a polynomial asymptotic stability.In fact, he considered the kernels having small flat zones and these zones are not too big (see also [24] for the case of coupled system).More recently, under the assumptions that [1] genera- lized these works to an arbitrary decay for wave equation with a viscoelastic damping term.Moreover, we would like to mention some results in [25]- [30].
The rest of our paper is organized as follows.In Section 2, we give some preparations for our consideration and our main result.The statements and the proofs of our main results will be given in Section 3.
For convenience, we denote the norm and scalar product in

Preliminaries and Main Result
For the memory kernel g we assume that: ( ) H suppose that there exists a nondecreasing function ( ) For the functions p and q , we assume that ( ) 0 , p q C ∈ Γ and ( ) 0 p x > and ( ) 0 q x > for all 0 x ∈ Γ .This assumption implies that there exist positive constants ( ) We use the notation d , and ,  d .
Let λ and λ  be the smallest positive constants such that Firstly, we have the following existence and uniqueness results, it can be established by adopting the arguments of [2] [31].
hold.There exists a unique pair of functions ( ) , t u y , which is a solution to the problem (1.1) in the class .
We introduce the modified energy functional where To state our main result, we introduce the following notations as in [32].For every measurable set , we define the probability measure ĝ by ( ) ( ) The flatness set and the flatness rate of g are defined by ( ) ( ) { } : 0 and 0 g s g s g s (2.9) and ( ) ( ) respectively.We denote Now, we are in a position to state our main result.
hold and , then there exist positive constants C and ν such that

Arbitrary Rate of Decay
Now we define ( ) Using (1.1) and (3.1), we have We use here the following identity due to [1], to give a better estimate for the term ( ) ( ) From (2.1), (3.2) and (3.3), integration by parts and Young's inequality, we derive for any As in [5], we have: Lemma 3.1 For ( ) Now we define the functional It follows from (1.1) and (3.6) that   can be estimated as in [1]: where ĝ is defined in (2.8).For any Taking into account these estimates in (3.6), let t * be a number such that ( ) , we obtain that then we know from [1] that At the same time, we have the following lemmas.
Lemma 3.2 For M large enough, there exist two positive constants 1 ρ and For n ∈  , as in [32] we introduce the sets It is easy to see that where g  is given in (2.9) and g  is the null set where g′ is not defined.Additionally, we denote    for some positive constants C and ν .
of the present paper is to provide some results on the long-time behavior to this equation with acoustic boundary conditions.By using the assumptions on the relaxation function due to Tatar[1], we show an arbitrary rate of decay with not necessary of an exponential or polynomial one and without the as-

>
is a positive function.Similarly, Han and Wang[20] proved the energy decay for the viscoelastic equation with nonlinear damping are constants.Then Park and Park[21] established the general decay for the viscoelastic problem with nonlinear weak damping

( ) 2 L
Ω by ⋅ and ( ) , ⋅ ⋅ , respectively.C denotes a general positive constant, which may be different in different estimates.Z. Y. Ma

8 I
, for 3 4 , 0 δ δ > , we use a different estimate as (2.11), we define the following functional in (3.18), it follows that