Asymptotic Behaviour to a Von Kármán System with Internal Damping ()
1. Introduction
Theodor von Kármán (1910) [2] started the nonlinear system of partial differential for great deflections and for the Airy stress function of a thin elastic plate. For several years this system was studied in different situations. Using frictional dissipation at boundary, I. Lasiecka et al. [3-5] proved the uniform decay of the solution. G. P. Menzala and E. Zuazua [6] by semigroup properties gave the exponential decay when thermal damping was considered. For Viscoelastic plates with memory, J. E. M. Rivera et al. [7,8] proved that the energy decays uniformly, exponentially or algebraically with the same rate of decay of the relaxation function. C. A. Raposo and M. L. Santos [9] gave a General Decay of solution for the memory case. In [10-13] the authors consider the von Kármán system with frictional dissipations effective in the whole plate, in a part of the plate or at the boundary. It is shown in these works that these dissipations produce uniform rate of decay of the solution when t goes to infinity. In this work we also consider the system with internal damping, which is the natural problem. A distinctive feature of our paper is to use Nakao’s method to show that the energy decays exponentially to zero.
2. Existence of Solution
We use the standard Lebesgue space and Sobolev space with their usual properties as in [14] and in this sense 
and 
denotes the inner product in 
and 
respectively and by 
we denote the usual norm in
. Let 
be a bounded domain of the plane with regular boundary
. For a real number 
we denote 
and
. Here 
is the displacement, 
the Airy stress function and 
is the unit normal external in
. With this notation we have the following system
(1)
(2)
(3)
(4)
where
Now using the same idea of [6] we have the following result of existence of solution.
Theorem 2.1. For 
there exists 
such that
weak solution of (1)-(4).
Proof. We defining the energy 
of the system (1)-(4) by
.
This system is well posed in the energy space (see [15]) and we have and E’(t) < 0. Galerkin’s method together with the dissipative properties of the energy give us global existence of solution in the energy space. Finally using the results from [5] on the regularity properties of von Kármám bracket the uniqueness follows.
3. Asymptotic Behaviour
In this section, we will use the Theorem of Nakao to prove the exponential decay of the solution.
Theorem 3.1. (Theorem of Nakao) Let 
be a nonnegative function on 
satisfying
where 
is a positive constant. Then we have
.
Proof. See page 748 of [1].
In the sequel we have two lemmasLemma 3.1. The functional 
satisfies
.
Proof. Multiplying (1) by 
and integrating in
, we have
Using (2) we obtain
from where follows
(5)
Performing integration in
, we have
(6)
then
. (7)
Lemma 3.2. The functional
satisfies
.
Proof. First we note that
(8)
From (7) we get 
and 
such that
. (9)
Multiplying (1) by u and integrating in
, we have
Integrating from 
to 
and using (8) we have
Now, choosing C such that 
and applying Cauchy-Schuwarz inequality we get
and using (9),
from where follows
. (10)
Now we are in position of to prove our principal result.
Theorem 3.2. The solution 
satisfies
(11)
for almost every
, with
, constants independents from t.
Proof. From (7) and (10) we obtain
.
There exists 
such that
(12)
From (6) we get
.
Then
and
Now using (11) and (12) we obtain
then
and finally by Theorem of Nakao follows
with
.
NOTES