Abstract
In this paper we consider Kirchhoff plates with a memory condition at the boundary. For a wider class of relaxation functions, we establish a more general decay result, from which the usual exponential and polynomial decay rates are only special cases.
MSC: 35B40, 74K20, 35L70.
Keywords:
Kirchhoff plates; general decay rate; memory term; relaxation function1 Introduction
We consider the following Kirchhoff plates with a memory condition at the boundary:
where
Let us denote by
where
and the constant μ,
In (1.1), u denotes the position of the plate. The integral equations (1.3) and (1.4) describe
the memory effects which can be caused, for example, by the interaction with another
viscoelastic element. The relaxation functions
If we denote the compactness of
The uniform stabilization of Kirchhoff plates with linear or nonlinear boundary feedback was investigated by several authors; see, for example, [13] among others. The uniform decay for plates with memory was studied in [46] and the references therein. There exists a large body of literature regarding viscoelastic problems with the memory term acting in the domain or at the boundary (see [712]). Rivera and Racke [13] investigated the decay results for magnetothermoelastic system. Santos et al.[14] studied the asymptotic behavior of the solutions of a nonlinear wave equation of Kirchhoff type with a boundary condition of memory type. Cavalcanti and Guesmia [15] proved the general decay rates of solutions to a nonlinear wave equation with a boundary condition of memory type. Park and Kang [16] studied the exponential decay for the multivalued hyperbolic differential inclusion with a boundary condition of memory type. Kafini [17] showed the decay results for viscoelastic diffusion equations in the absence of instantaneous elasticity. They proved that the energy decays uniformly exponentially or algebraically at the same rate as the relaxation functions. In the present work, we generalize the earlier decay results of the solution of (1.1)(1.5). More precisely, we show that the energy decays at the rate similar to the relaxation functions, which are not necessarily decaying like polynomial or exponential functions. In fact, our result allows a larger class of relaxation functions. Recently, Messaoudi and Soufyane [18], Santos and Soufyane [19], and Mustafa and Messaoudi [20] proved the general decay for the wave equation, von Karman plate system, and Timoshenko system with viscoelastic boundary conditions, respectively.
The paper is organized as follows. In Section 2 we present some notations and material needed for our work. In Section 3 we prove the general decay of the solutions to the Kirchhoff plates with a memory condition at the boundary.
2 Preliminaries
In this section, we present some material needed in the proof of our main result. We use the standard Lebesgue and Sobolev spaces with their usual scalar products and norms. Define the following space:
First, we shall use Eqs. (1.3) and (1.4) to estimate the values
the convolution product operator and differentiating Eqs. (1.3) and (1.4), we arrive at the following Volterra equations:
Applying the Volterra inverse operator, we get
where the resolvent kernels of
Denoting by
Therefore, we use (2.1) and (2.2) instead of the boundary conditions (1.3) and (1.4).
Let us define the bilinear form
We state the following lemma which will be useful in what follows.
Lemma 2.1 ([2])
Letuandvbe functions in
Let us denote
The following lemma states an important property of the convolution operator.
Lemma 2.2For
The proof of this lemma follows by differentiating the term
We formulate the following assumption:
(A1) Let
Let us introduce the energy function
In these conditions, we are able to prove the existence of a strong solution.
Theorem 2.1Let
If
then there is only one solutionuof the system (1.1)(1.5) satisfying
Proof See Park and Kang [16]. □
3 General decay
In this section, we show that the solution of the system (1.1)(1.5) may have a general decay not necessarily of exponential or polynomial type. For this we consider that the resolvent kernels satisfy the following hypothesis:
(H)
and there exists a nonincreasing continuous function
The following identity will be used later.
Lemma 3.1 ([2])
For every
Our point of departure will be to establish some inequalities for the strong solution of the system (1.1)(1.5).
Lemma 3.2The energy functionalEsatisfies, along the solution of (1.1)(1.5), the estimate
Proof Multiplying Eq. (1.1) by
Substituting the boundary terms by (2.1) and (2.2) and using Lemma 2.2 and the Young inequality, our conclusion follows. □
Let us consider the following binary operator:
Then applying the Holder inequality for
Let us define the functional
The following lemma plays an important role in the construction of the desired functional.
Lemma 3.3Suppose that the initial data
Proof Differentiating ψ, using Eq. (1.1), and taking
Let us next examine the integrals over
since
Therefore, from (3.4) and (3.5), we have
Using the Young inequality, we have
where ϵ is a positive constant. Since the bilinear form
where
where
Since the boundary conditions (2.1) and (2.2) can be written as
our conclusion follows. □
Let us introduce the Lyapunov functional
with
Theorem 3.1Let
Otherwise,
for all
and
Proof Applying the inequality (3.3) with
Then, choosing N large enough and
On the other hand, we can choose N even larger so that
If
which gives
Using the fact that ξ is a nonincreasing continuous function as
Since using (3.14),
we obtain for some positive constant ω,
Case 1: If
A simple integration over
By using (3.2) and (3.15), we then obtain for some positive constant C
Thus, the estimate (3.11) is proved.
Case 2: If
where
In this case, we introduce
A simple differentiation of G, using (3.17), leads to
Again, a simple integration over
which implies, for all
By using (3.15), we deduce
Consequently, by the boundedness of ξ, (3.12) is established. □
Remark 3.1 Estimates (3.11) and (3.12) are also true for
Remark 3.2 Note that the exponential and polynomial decay estimates are only particular cases
of (3.11) and (3.12). More precisely, we have exponential decay for
Example 3.1 As in [20], we give some examples to illustrate the energy decay rates given by (3.11).
(1) If
(2) If
The above two examples are included in the following more general one.
(3) For any nonincreasing functions
Competing interests
The author declares that she has no competing interests.
Author’s contributions
The work was realized by the author.
Acknowledgements
The author thanks the anonymous referee for a careful review. This work was supported by the DongA University research fund.
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