Asymptotic Behaviour to a Von Kármán System with Internal Damping

Abstract

In this work we consider the Von Kármán system with internal damping acting on the displacement of the plate and using the Theorem due to Nakao [1] we prove the exponential decay of the solution.

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Pereira, D. , Raposo, C. and Maranhão, C. (2012) Asymptotic Behaviour to a Von Kármán System with Internal Damping. Applied Mathematics, 3, 210-212. doi: 10.4236/am.2012.33033.

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

Conflicts of Interest

The authors declare no conflicts of interest.

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