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 function
We consider the following Kirchhoff plates with a memory condition at the boundary:
where and Ω is an open bounded set of with a regular boundary Γ. We divide the boundary into two parts:
Let us denote by the external unit normal to Γ, and let us denote by the unit tangent positively oriented on Γ. We are denoting by , the following differential operators:
where and are given by
and the constant μ, , represents Poisson’s ratio.
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 are positive and nondecreasing. This system models the small transversal vibrations of a thin plate whose Poisson coefficient is equal to μ. We assume that there exists such that
If we denote the compactness of by , the condition (1.7) implies that there exists a small positive constant such that , .
The uniform stabilization of Kirchhoff plates with linear or nonlinear boundary feedback was investigated by several authors; see, for example, [1-3] among others. The uniform decay for plates with memory was studied in [4-6] 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 [7-12]). Rivera and Racke  investigated the decay results for magneto-thermo-elastic system. Santos et al. 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  proved the general decay rates of solutions to a nonlinear wave equation with a boundary condition of memory type. Park and Kang  studied the exponential decay for the multi-valued hyperbolic differential inclusion with a boundary condition of memory type. Kafini  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 , Santos and Soufyane , and Mustafa and Messaoudi  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.
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 and on . Denoting by
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 satisfy
Denoting by , for , we obtain
Therefore, we use (2.1) and (2.2) instead of the boundary conditions (1.3) and (1.4).
Let us define the bilinear form as follows:
We state the following lemma which will be useful in what follows.
Lemma 2.1 ()
Letuandvbe functions in . Then we have
Let us denote
The following lemma states an important property of the convolution operator.
Lemma 2.2For , we have
The proof of this lemma follows by differentiating the term .
We formulate the following assumption:
(A1) Let satisfy in Ω for some .
Let us introduce the energy function
In these conditions, we are able to prove the existence of a strong solution.
Theorem 2.1Let be such that
If satisfy the compatibility condition
then there is only one solutionuof the system (1.1)-(1.5) satisfying
Proof See Park and Kang . □
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) is a twice differentiable function such that
and there exists a nonincreasing continuous function satisfying
The following identity will be used later.
Lemma 3.1 ()
For every and for every , we have
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 , integrating over Ω, and using Lemma 2.1, we get
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 , we have
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 and satisfies the compatibility condition (2.5). Then the solution of the system (1.1)-(1.5) satisfies
Proof Differentiating ψ, using Eq. (1.1), and taking in Lemma 3.1, we get
Let us next examine the integrals over in (3.4). Since on , we have on and
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 is strictly coercive on W, using the trace theory, we obtain
where is a constant depending on Ω and μ. Further, one has
where with some constant . Substituting the inequalities (3.7)-(3.10) into (3.6) and taking into account the fact that on , we have
Since the boundary conditions (2.1) and (2.2) can be written as
our conclusion follows. □
Let us introduce the Lyapunov functional
with . Now we are in a position to show the main result of this paper.
Theorem 3.1Let . Suppose the resolvent kernels , satisfy the conditions (H) and (3.1). Then there exist constants such that for some large enough, the solution of (1.1)-(1.5) satisfies
for all , where
Proof Applying the inequality (3.3) with in Lemma 3.3 and from Lemma 3.2, we obtain
Then, choosing N large enough and , we obtain
On the other hand, we can choose N even larger so that
If , , then using (3.1) and (3.2), we have
Using the fact that ξ is a nonincreasing continuous function as and are nonincreasing, and so ξ is differentiable, with , for a.e. t, then we infer that
Since using (3.14),
we obtain for some positive constant ω,
Case 1: If on , then (3.16) reduces to
A simple integration over yields
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 on , then (3.16) gives
In this case, we introduce
A simple differentiation of G, using (3.17), leads to
Again, a simple integration over yields
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 by virtue of continuity and boundedness of and ξ.
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 and and polynomial decay for and , where and are positive constants.
Example 3.1 As in , we give some examples to illustrate the energy decay rates given by (3.11).
(1) If , , then for , , where . For suitably chosen positive constants a and b, satisfies (H) and (3.11) gives
(2) If , , and , , then for , , where . Then
The above two examples are included in the following more general one.
(3) For any nonincreasing functions , , which satisfy (H), are also nonincreasing differentiable functions, and , for some , (3.11) gives
The author declares that she has no competing interests.
The work was realized by the author.
The author thanks the anonymous referee for a careful review. This work was supported by the Dong-A University research fund.
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