Introduction

In this paper we deal with the Mindlin-Timoshenko plate system. We consider a plate of small thickness $h$ occupying a region $\Omega\subset {\mathbb{R}}^2$. Denoting by $\psi(x,y,t)$ and $\varphi(x,y,t)$ the angles of rotation of a filament and by $\omega(x,y,t)$ the vertical displacement of the middle surface, the equations (see [3]) that model this system are given by

$$\frac{\rho h^3}{12} \psi_{tt} -D\left( { \frac{\partial^2 {\psi}}... ... y} } \right) +K\left( {\psi + \frac{\partial {\omega}}{\partial x} } \right)=0 \Omega\times{\mathbb{R}}^+,$$ (1.1) in

$$\frac{\rho h^3}{12}\varphi_{tt} - D\left( { \frac{\partial^2 {\va... ... } \right) +K\left( {\varphi + \frac{\partial {\omega}}{\partial y} } \right)=0 \Omega\times{\mathbb{R}}^+,$$ (1.2) in

$$\rho h \omega_{tt} - K\left[ { \frac{\partial }{\partial x}\left(... ... y}\left( {\varphi + \frac{\partial {\omega}}{\partial y} } \right) } \right]=0 \Omega\times{\mathbb{R}}^+.$$ (1.3) in

Here, the positive constants $\rho,\ D,\ \mu$ and $K$ denote

$$\begin{array}{rcl} \rho &:& \mbox{the mass density per unit ... ...}\ (0<\mu<1/2), \\ K &:& \mbox{the shear modulus}. \end{array}$$

The boundary $\Gamma$ of $\Omega$ is a smooth surface composed by two components $\Gamma_0$, $\Gamma_1$ ( $\Gamma=\Gamma_0\cup\Gamma_1$) such that $\bar{\Gamma}_0\cap \bar{\Gamma}_1 = \emptyset$.

We assume that the plate is clamped along $\Gamma_0$, that is

$$\parbox{0cm}{ \begin{Beqnarray*}\psi=\varphi=\omega=0 &\mbox{on}& \Gamma_0\times {\mathbb R}^+, \end{Beqnarray*} }$$ (1.4)

and the other part of its boundary $\Gamma_1$ is in connection with a viscoelastic element, which produces the following boundary condition

$$\parbox{0cm}{ \begin{Beqnarray*}\psi + \int_0^t{g_1(t-s)\mathfrak... ...omega(s)) }\, ds=0 &\mbox{on}& \Gamma_1\times {\mathbb R}^+, \end{Beqnarray*} }$$ (1.5)

where the boundary operators $\mathfrak{B}_1$, $\mathfrak{B}_2$ and $\mathfrak{B}_3$ are given by

$$\mathfrak{B}_1(\psi,\varphi) = D\left[ { \nu_1\frac{\partial {\psi}}{\partial x}+\mu\nu_1\frac{\... ...i}}{\partial y}+\frac{\partial {\varphi}}{\partial x}} \right)\nu_2 } \right] ,$$

$$\mathfrak{B}_2(\psi,\varphi) = D\left[ { \nu_2\frac{\partial {\varphi}}{\partial y}+\mu\nu_2\fra... ...i}}{\partial y}+\frac{\partial {\varphi}}{\partial x}} \right)\nu_1 } \right] ,$$

$$\mathfrak{B}_3(\psi,\varphi,\omega) = K\left( { \frac{\partial \omega}{\partial \nu}+\nu_1\psi +\nu_2\varphi} \right),$$

and $g_i$, $i=1,2,3$ are non negative decreasing functions. The initial position for $\psi, \varphi$ and $\omega$ are prescribed by

$$\begin{array}{r} \psi(x,0)=\psi^0(x),\quad \varphi(x,0)=\varp... ...x,0) =\omega^1(x) \end{array} \quad \quad\mbox{in}\quad \Omega.$$ (1.6)

where $\psi^0$, $\psi^1$, $\varphi^0$, $\varphi^1$, $\omega^0$ and $\omega^1$ are known functions. A typical example of $\Omega$ is given in the next figure

Concerning memory condition on the boundary we can cite only a few works. In [1] Ciarletta established theorems of existence, uniqueness and asymptotic stability for a linear model of heat conduction. In this case the memory condition describes a boundary that can absorb heat and due to the hereditary term, can retain part of it. In [2] Fabrizio and Morro consider a linear electromagnetic model and proved the existence, uniqueness and asymptotic stability of the solutions. In [6] Muñoz Rivera and Andrade showed exponential stability for a non homogeneous anisotropic system when the resolvent kernel of the memory is exponential type. Polynomial resolvent kernel was not considered in that work.

The nonlinear one-dimensional wave equation with memory condition on the boundary was studied by Qin [7], he showed existence, uniqueness and stability of global solutions provided the initial data is small in $H^3\times H^2$, this result was improved by Muñoz Rivera and Andrade [5], by taking small initial data in $H^2\times H^1$.

The uniform stabilization of system (1.1)-(1.4) for frictional boundary conditions instead of condition (1.5) was studied by Lagnese [3], who proved, under some geometrical conditions that the energy associated to the Mindlin-Timoshenko's plate decays exponentially as time goes to infinity.

The aim of this paper is to study the asymptotic behavior of solutions of system (1.1)-(1.6). We show that the solution decays exponentially to zero provided the relaxation functions $g_i$ decays exponentially to zero. Moreover, if $g_i$ decays polynomially, then we show that the corresponding solution also decays polynomially to zero with the same rate of decay.

The remainder part of this paper is organized as follows. In the next section we establish the existence, uniqueness and regularity of solutions. In section 3 we show the exponential decay of the first order energy and finally, in section 4 we prove the polynomial decay.