General decay for Kirchhoff plates with a boundary condition of memory type
© Kang; licensee Springer. 2012
Received: 30 July 2012
Accepted: 25 October 2012
Published: 7 November 2012
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.
and the constant μ, , represents Poisson’s ratio.
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.
Therefore, we use (2.1) and (2.2) instead of the boundary conditions (1.3) and (1.4).
We state the following lemma which will be useful in what follows.
Lemma 2.1 ()
The following lemma states an important property of the convolution operator.
The proof of this lemma follows by differentiating the term .
We formulate the following assumption:
(A1) Let satisfy in Ω for some .
In these conditions, we are able to prove the existence of a strong solution.
Proof See Park and Kang . □
3 General decay
- (H)is a twice differentiable function such that
The following identity will be used later.
Lemma 3.1 ()
Our point of departure will be to establish some inequalities for the strong solution of the system (1.1)-(1.5).
Substituting the boundary terms by (2.1) and (2.2) and using Lemma 2.2 and the Young inequality, our conclusion follows. □
The following lemma plays an important role in the construction of the desired functional.
our conclusion follows. □
with . Now we are in a position to show the main result of this paper.
Thus, the estimate (3.11) is proved.
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.
- (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
- (3)For any nonincreasing functions , , which satisfy (H), are also nonincreasing differentiable functions, and , for some , (3.11) gives
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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