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:
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
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:
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
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 ()
Let us denote
The following lemma states an important property of the convolution operator.
We formulate the following assumption:
Let us introduce the energy function
In these conditions, we are able to prove the existence of a strong solution.
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:
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).
Lemma 3.2The energy functionalEsatisfies, along the solution of (1.1)-(1.5), the estimate
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:
Let us define the functional
The following lemma plays an important role in the construction of the desired functional.
Therefore, from (3.4) and (3.5), we have
Using the Young inequality, we have
Since the boundary conditions (2.1) and (2.2) can be written as
our conclusion follows. □
Let us introduce the Lyapunov functional
On the other hand, we can choose N even larger so that
Since using (3.14),
we obtain for some positive constant ω,
By using (3.2) and (3.15), we then obtain for some positive constant C
Thus, the estimate (3.11) is proved.
In this case, we introduce
A simple differentiation of G, using (3.17), leads to
By using (3.15), we deduce
Consequently, by the boundedness of ξ, (3.12) is established. □
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).
The above two examples are included in the following more general one.
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.
Komornik, V: On the nonlinear boundary stabilization of Kirchhoff plates. Nonlinear Differ. Equ. Appl.. 1, 323–337 (1994). Publisher Full Text
Lasiecka, I: Exponential decay rates for the solutions of Euler-Bernoulli moments only. J. Differ. Equ.. 95, 169–182 (1992). Publisher Full Text
Munoz Rivera, JE, Lapa, EC, Barreto, R: Decay rates for viscoelastic plates with memory. J. Elast.. 44, 61–87 (1996). Publisher Full Text
Santos, ML, Junior, F: A boundary condition with memory for Kirchhoff plates equations. Appl. Math. Comput.. 148, 475–496 (2004). Publisher Full Text
Aassila, M, Cavalcanti, MM, Soriano, JA: Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain. SIAM J. Control Optim.. 38(5), 1581–1602 (2000). Publisher Full Text
Cavalcanti, MM, Domingos Cavalcanti, VN, Soriano, JA: Exponential decay for the solution of semilinear viscoelastic wave equation with localized damping. Electron. J. Differ. Equ.. 2002, (2002) Article ID 44
Cavalcanti, MM, Domingos Cavalcanti, VN, Martinez, P: General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal.. 68(1), 177–193 (2008). Publisher Full Text
Kang, JR: Energy decay rates for von Kármán system with memory and boundary feedback. Appl. Math. Comput.. 218, 9085–9094 (2012). Publisher Full Text
Messaoudi, SA: General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl.. 341, 1457–1467 (2008). Publisher Full Text
Messaoudi, SA, Tatar, NE: Exponential and polynomial decay for a quasilinear viscoelastic equation. Nonlinear Anal.. 68, 785–793 (2008). Publisher Full Text
Santos, ML, Ferreira, J, Pereira, DC, Raposo, CA: Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary. Nonlinear Anal.. 54, 959–976 (2003). Publisher Full Text
Park, JY, Kang, JR: A boundary condition with memory for the Kirchhoff plate equations with non-linear dissipation. Math. Methods Appl. Sci.. 29, 267–280 (2006). Publisher Full Text
Messaoudi, SA, Soufyane, A: General decay of solutions of a wave equation with a boundary control of memory type. Nonlinear Anal., Real World Appl.. 11, 2896–2904 (2010). Publisher Full Text
Mustafa, MI, Messaoudi, SA: Energy decay rates for a Timoshenko system with viscoelastic boundary conditions. Appl. Math. Comput.. 218, 9125–9131 (2012). Publisher Full Text