General decay for Kirchhoff plates with a boundary condition of memory type
Kang Boundary Value Problems 2012, 2012:129
http://www.boundaryvalueproblems.com/content/2012/1/129
RESEARCH
Open Access
General decay for Kirchhoff plates with a
boundary condition of memory type
Jum-Ran Kang*
*
Correspondence:
Department of Mathematics,
Dong-A University, Saha-Ku, Busan,
604-714, Korea
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 function
1 Introduction
We consider the following Kirchhoff plates with a memory condition at the boundary:
utt + u + a(x)ut = ,
in × (, ∞),
∂u
= , on × (, ∞),
∂ν
t
g (t – s)B u(s) ds = ,
–u +
(.)
(.)
u=
on × (, ∞),
(.)
on × (, ∞),
(.)
∂u
+
∂ν
t
g (t – s)B u(s) ds = ,
u(, x) = u (x),
ut (, x) = u (x),
in ,
(.)
¯ and is an open bounded set of R with a regular boundary . We divide
where a ∈ C ()
the boundary into two parts:
= ∪
with ¯ ∩ ¯ = ∅; and = ∅.
Let us denote by ν = (ν , ν ) the external unit normal to , and let us denote by η = (–ν , ν )
the unit tangent positively oriented on . We are denoting by B , B the following differential operators:
B u = u + ( – μ)B u,
B u =
∂u
+ ( – μ)B u,
∂ν
where B and B are given by
B u = ν ν uxy – ν uyy – ν uxx ,
B u =
∂
ν – ν uxy + ν ν (uyy – uxx ) ,
∂η
and the constant μ, < μ < , represents Poisson’s ratio.
© 2012 Kang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
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In (.), u denotes the position of the plate. The integral equations (.) and (.) describe
the memory effects which can be caused, for example, by the interaction with another viscoelastic element. The relaxation functions g , g ∈ C (, ∞) 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 x ∈ R such that
= x ∈ : ν(x) · (x – x ) ≤ ,
= x ∈ : ν(x) · (x – x ) > .
(.)
(.)
If we denote the compactness of by m(x) = x – x , the condition (.) implies that there
exists a small positive constant δ such that < δ ≤ m(x) · ν(x), ∀x ∈ .
The uniform stabilization of Kirchhoff plates with linear or nonlinear boundary feedback was investigated by several authors; see, for example, [–] among others. The uniform decay for plates with memory was studied in [–] 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 [–]). 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 (.)-(.). 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 we present some notations and material needed for our work. In Section 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:
W = w ∈ H (); w =
∂w
= on .
∂ν
First, we shall use Eqs. (.) and (.) to estimate the values B and B on . Denoting by
(g ∗ v)(t) =
t
g(t – s)v(s) ds,
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the convolution product operator and differentiating Eqs. (.) and (.), we arrive at the
following Volterra equations:
B u +
g ∗ B u =
ut ,
g ()
g ()
B u +
∂ut
g ∗ B u = –
.
g ()
g () ∂ν
Applying the Volterra inverse operator, we get
B u =
{ut + k ∗ ut },
g ()
B u = –
∂ut
∂ut
+ k ∗
,
g () ∂ν
∂ν
g
i
satisfy
where the resolvent kernels of – gi ()
ki +
gi ∗ ki = –
g,
gi ()
gi () i
∀i = , .
Denoting by τi = gi ()
, for i = , , we obtain
B u = τ ut + k ()u – k (t)u + k ∗ u ,
(.)
∂u
∂u
∂u
∂ut
+ k ()
– k (t)
+ k ∗
.
∂ν
∂ν
∂ν
∂ν
B u = –τ
(.)
Therefore, we use (.) and (.) instead of the boundary conditions (.) and (.).
Let us define the bilinear form a(·, ·) as follows:
a(u, v) =
uxx vxx + uyy vyy + μ(uxx vyy + uyy vxx ) + ( – μ)uxy vxy dx dy.
(.)
We state the following lemma which will be useful in what follows.
Lemma . ([]) Let u and v be functions in H () ∩ W . Then we have
u v dx = a(u, v) +
(B u)v – (B u)
∂v
d.
∂ν
(.)
Let us denote
(g v)(t) :=
t
g(t – s) v(t) – v(s) ds.
The following lemma states an important property of the convolution operator.
Lemma . For g, v ∈ C ([, ∞) : R), we have
d
(g ∗ v)vt = – g(t) v(t) + g v –
g v –
dt
t
g(s) ds |v| .
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The proof of this lemma follows by differentiating the term g v.
We formulate the following assumption:
¯ satisfy a(x) ≥ a > in for some a .
(A) Let a ∈ C ()
Let us introduce the energy function
E(t) =
|ut | dx + a(u, u) + τ
(...truncated)
