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.

Von Kármán System; Internal Damping; Exponential Decay; Theorem of Nakao1. 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

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

Performing integration in, we have

then

Lemma 3.2. The functional

satisfies

.

Proof. First we note that

From (7) we get and such that

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

Now we are in position of to prove our principal result.

Theorem 3.2. The solution satisfies

for almost every, with, constants independents from t.

Proof. From (7) and (10) we obtain

.

There exists such that

From (6) we get

.

Then

and

Now using (11) and (12) we obtain

then

and finally by Theorem of Nakao follows

with.

REFERENCESNOTESReferencesM. Nakao, “A Difference Inequalit and Its Application to Nonolinear Evolution Equation,” Journal of the Mathematical Society of Japan, Vol. 30, No. 4, 1978, pp. 747-762. doi:10.2969/jmsj/03040747T. V. Kármán, “Festigkeitsprobleme im Maschinenbaum. Encyklopadie der Math,” Wiss. V/4C, Leipzig, 1910, pp. 311-385.M. Horn and I. Lasiecka, “Uniform Decay of Weak Solutions to a Von Kármán Plate with Nonlinear Boundary Dissipation,” Differential and Integral Equations, Vol. 7, No. 4, 1994, pp. 885-908.M. Horn and I. Lasiecka, “Global Stabilization of a Dynamical Von Kármán Plate with Nonlinear Boundary Feedback,” Applied Mathematics & Optimization, Vol. 31, No. 1, 1995, pp. 57-84. doi:10.1007/BF01182557M. Horn, A. Favini, I. Lasiecka and D. Tataru, “Global Existence, Uniqueness and Regularity to a Von Kármán System with Nonlinear Boundary Dissipation,” Differential and Integral Equations, Vol. 9, No. 2, 1996, pp. 267-294.G. P. Menzala and E. Zuazua, “Energy Decay Rates for the Von Kármán System of Thermoelastic Plates,” Differential and Integral Equations, Vol. 11, No. 5, 1998, pp. 755-770.J. E. M. Rivera and G. P. Menzala, “Decay Rates of Solutions of a Von Kármán System for Viscoelastic Plates with Memory,” Quarterly of Applied Mathematics, Estados Unidos, Vol. 82, No. 1, 1999, pp.181-200.J. E. M. Rivera, H. P. Oquendo and M. L. Santos, “Asymptotic Behavior to a Von Kármán Plate with Boundary Memory Conditions,” Nonlinear Analysis, Vol. 62, No. 7, 2005, pp. 1183-1205. doi:10.1016/j.na.2005.04.025C. A. Raposo and M. L. Santos, “General Decay to a Von Kármán System with Memory,” Nonlinear Analysis, Vol. 74, No. 3, 2011, pp. 937 945.
doi:10.1016/j.na.2010.09.047G. Avalos and I. Lasiecka, “Uniform Decays in Nonlinear Thermoelastic System,” In: W. Hager and P. Pardalos, Eds., Optimal Control, Theory Algorithms and Application, Kluwer, 1998, pp. 1-23.G. Avalos, I. Lasiecka and R. Triggiani, “Uniform Stability of Nonlinear Thermoelastic Plates with Free Boundary Conditions,” International Series of Numerical Mathematics, Vol. 133, 1999, pp. 13-32.H. Koch and A. Stahel, “Global Existence of Classical Solutions to the Dynamical Von Kármán Equations,” Mathematical Methods in the Applied Sciences, Vol. 161, 1993, pp. 581-586. doi:10.1002/mma.1670160806J. Puel and M. Tucsnak, “Boundary Stabilization for the Von Kármán Equations,” SIAM Journal on Control and Optimization, Vol. 33, No. 1, 1996, pp. 255-273.
doi:10.1137/S0363012992228350A. Adams, “Sobolev Spaces,” Academic Press, New York, 1975.G. P. Menzala, V. Bisognin, E. Bisognin and E. Zuazua, “On Exponential Stability for Von Kármán Equations in the Presence of Thermal Effects,” Mathematical Methods in the Applied Sciences, Vol. 21, No. 5, 1988, pp. 393-416.