Decay Rate for a Viscoelastic Equation with Strong Damping and Acoustic Boundary Conditions ()
1. Introduction
In this paper, we investigate the following viscoelastic system with acoustic boundary conditons
(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)
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
(1.7)
Our problem is of the form
(1.8)
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 [3] , 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 [4] for several other related models.
Recently, Liu [5] considered the following viscoelastic problem with acoustic boundary conditions
(1.9)
(1.10)
(1.11)
(1.12)
(1.13)
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
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
. 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 [9] investigated problems (1.9)-(1.13) and proved general rates of decay which depended on the behavior of
, under the additional assumption of
that
Many authors have focused on the viscoelastic problem. In the pioneer work of Dafermos [11] [12] , 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 [13] [14] [15] . 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. [2] studied the following equation with Dirichlet boundary conditions
(1.14)
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 [16] established, for small initial data, the global existence and uniform stability of solutions to the equation
(1.15)
with Dirichlet boundary condition, where
are constants. In the case
in (15), 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
, where
is a positive function. Similarly, Han and Wang [20] proved the energy decay for the viscoelastic equation with nonlinear damping
(1.16)
with Dirichlet boundary condition, where
are constants. Then Park and Park [21] established the general decay for the viscoelastic problem with nonlinear weak damping
(1.17)
with the Dirichlet boundary condition, where
is a constant. We also mention that Fabrizio and Polidoro [22] obtained the exponential decay result under the conditions that
and
for some
. 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
and
for some nonnegative function
, Tatar [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 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
(2.1)
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
(2.2)
We use the notation
Let
and
be the smallest positive constants such that
(2.3)
Firstly, we have the following existence and uniqueness results, it can be established by adopting the arguments of [2] [31] .
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
(2.4)
(2.5)
We introduce the modified energy functional
(2.6)
where
Clearly
(2.7)
To state our main result, we introduce the following notations as in [32] . For every measurable set
, we define the probability measure
by
(2.8)
The flatness set and the flatness rate of
are defined by
(2.9)
and
(2.10)
respectively. We denote
(2.11)
Now, we are in a position to state our main result.
Theorem 2.2 ( [23] ) Let
, Assume that (2.1)-(2.2) hold and
. If
, then there exist positive constants
and
such that
(2.12)
3. Arbitrary Rate of Decay
Now we define
(3.1)
Using (1.1) and (3.1), we have
(3.2)
We use here the following identity due to [1] , to give a better estimate for the
term
:
(3.3)
From (2.1), (3.2) and (3.3), integration by parts and Young’s inequality, we derive for any
,
(3.4)
As in [5] , we have:
Lemma 3.1 For
, we have
(3.5)
Now we define the functional
(3.6)
It follows from (1.1) and (3.6) that
(3.7)
For any
, we have
(3.8)
For all measurable sets
and
such that
,
,
and
can be estimated as in [1] :
(3.9)
(3.10)
(3.11)
where
is defined in (2.8). For any
,
(3.12)
For
, for
, we use a different estimate as
(3.13)
Taking into account these estimates in (3.6), let
be a number such that
, we obtain that
(3.14)
Let
(3.15)
and
is given in (2.11), we define the following functional
(3.16)
then we know from [1] that
(3.17)
At the same time, we have the following lemmas.
Lemma 3.2 For
large enough, there exist two positive constants
and
such that
(3.18)
Proof. See, e.g. Liu [5] .
Proof of Theorem 2.2 By using (2.7), (3.4), (3.13)-(3.16), a series of com- putations yields, for
,
(3.19)
For
, as in [32] we introduce the sets
(3.20)
It is easy to see that
(3.21)
where
is given in (2.9) and
is the null set where
is not defined. Additionally, we denote
, then
(3.22)
since
for all
and
. Then, we take
and
in (3.18), it follows that
(3.23)
for some
. Since
, we can choose
small enough
and
large enough such that
(3.24)
and
(3.25)
with
. Note that for
large enough. Furthermore, we
require that
(3.26)
Combining (3.24) and (3.25), we obtain
(3.27)
Choose our constants properly so that:
(3.28)
(3.29)
(3.30)
together with (3.22) yield
(3.31)
As
is decreasing, we have
for all
. Then (3.30) becomes
Since
is equipped with
, we get
(3.32)
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
(3.33)
for some positive constants
and
.
Acknowledgements
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.