In this paper, we investigate the following viscoelastic system with acoustic boundary conditons
where is a bounded domain with smooth boundary , is the unit outward normal to , the function represents the kernel of a memory, and are specific functions, and is a real number such that
Our problem is of the form
which has several modeling features. In the case, is a constant; Equation (8) has been used to model extensional vibrations of thin rods (see Love  , Chapter 20). In the case, is not a constant; Equation (8) can model materials whose density depends on the velocity , 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  for several other related models.
Recently, Liu  considered the following viscoelastic problem with acoustic boundary conditions
the authors obtain an arbitrary decay rate of the energy. In the pioneering paper  , 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 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     and references therein. For instance, Rivera and Qin  proved the polynomial decay for the wave motion with general acoustic boundary conditions by using the Lyapunov functional technique. Frota and Larkin  established global solvability and the exponential decay for problems (1.9)-(1.13) with . They overcame the difficulties which were arisen due to the absence of in (1.12) by using the degenerated second order equation. Recently, Park and Park  investigated problems (1.9)-(1.13) and proved general rates of decay which depended on the behavior of , under the additional assumption of
Many authors have focused on the viscoelastic problem. In the pioneer work of Dafermos   , existence and asymptotic stability for a one-dimensional viscoelastic problem were proved but no rate of decay has been specified. Since then problems related to viscoelasticity have attracted a great deal of attention    . It seems all started with kernels of the form , then with kernels satisfying , for all , for some constants and and some other conditions on the second derivative, Cavalcanti et al.  studied the following equation with Dirichlet boundary conditions
where . They established a global existence result for and an exponential decay of energy for , and studied the interaction within the and the memory term . Messaoudi and Tatar  established, for small initial data, the global existence and uniform stability of solutions to the equation
with Dirichlet boundary condition, where are constants. In the case in (15), Messaoudi and Tatar  proved the exponential decay of global solutions to (15) without smallness of initial data, considering only the dissipation effect given by the memory.
In   , the condition has been replaced by , where is a positive function. Similarly, Han and Wang  proved the energy decay for the viscoelastic equation with nonlinear damping
with Dirichlet boundary condition, where are constants. Then Park and Park  established the general decay for the viscoelastic problem with nonlinear weak damping
with the Dirichlet boundary condition, where is a constant. We also mention that Fabrizio and Polidoro  obtained the exponential decay result under the conditions that and for some . Recently, Tatar  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  for the case of coupled system). More recently, under the assumptions that and for some nonnegative function , Tatar  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  -  .
The rest of our paper is organized as follows. In Section 2, we give some pre- parations 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 by and , respectively. denotes a general positive constant, which may be different in different estimates.
2. Preliminaries and Main Result
For the memory kernel we assume that:
is a non-increasing differentiable function satisfying that
suppose that there exists a nondecreasing function such
that is a decreasing function and .
For the functions and , we assume that and and for all . This assumption implies that there exist positive constants such that
We use the notation
Let and 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   .
Theorem 2.1 Let . Assume that and (2.2) hold. There exists a unique pair of functions , which is a solution to the problem (1.1) in the class
We introduce the modified energy functional
To state our main result, we introduce the following notations as in  . For every measurable set , we define the probability measure by
The flatness set and the flatness rate of are defined by
respectively. We denote
Now, we are in a position to state our main result.
Theorem 2.2 (  ) Let , Assume that (2.1)-(2.2) hold and . If , then there exist positive constants
and such that
3. Arbitrary Rate of Decay
Now we define
Using (1.1) and (3.1), we have
We use here the following identity due to  , to give a better estimate for the
From (2.1), (3.2) and (3.3), integration by parts and Young’s inequality, we derive for any ,
As in  , we have:
Lemma 3.1 For , we have
Now we define the functional
It follows from (1.1) and (3.6) that
For any , we have
For all measurable sets and such that , , and can be estimated as in  :
where is defined in (2.8). For any ,
For , for , we use a different estimate as
Taking into account these estimates in (3.6), let be a number such that
, we obtain that
and is given in (2.11), we define the following functional
then we know from  that
At the same time, we have the following lemmas.
Lemma 3.2 For large enough, there exist two positive constants and such that
Proof. See, e.g. Liu  .
Proof of Theorem 2.2 By using (2.7), (3.4), (3.13)-(3.16), a series of com- putations yields, for ,
For , as in  we introduce the sets
It is easy to see that
where is given in (2.9) and is the null set where is not defined. Additionally, we denote , then
since for all and . Then, we take and in (3.18), it follows that
for some . Since , we can choose small enough
and large enough such that
with . Note that for large enough. Furthermore, we
Combining (3.24) and (3.25), we obtain
Choose our constants properly so that:
together with (3.22) yield
As is decreasing, we have for all . Then (3.30) becomes
Since is equipped with , we get
integrating (3.31) over yields
Then using the left hand side inequality in (3.17), we get
By virtue of the continuity and boundedness of in the interval , we conclude that
for some positive constants and .
This work was in part supported by Shanghai Second Polytechnical University and the key discipline “Applied Mathematics” of Shanghai Second Polytechnic University with contract number XXKZD1304.
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