This paper is concerned with the existence and decay of solutions of the following Timoshenko system: $$ \left\|\begin{ array}{ cc} u"-\mu(t)\Delta u+\alpha_1 \displaystyle\sum_{ i=1}^{ n}\frac{ \partial v}{ \partial x_{ i}}=0,\, \in \Omega\times (0, \infty),\\ v"-\Delta v-\alpha_2 \displaystyle\sum_{ i=1}^{ n}\frac{ \partial u}{ \partial x_{ i}}=0, \, \in \Omega\times (0, \infty), \end{ array} \right. $$ subject to the nonlinear boundary conditions, $$ \left\|\begin{ array}{ cc} u=v=0 \,\, in \,\Gamma_{ 0}\times (0, \infty),\\ \frac{ \partial u}{ \partial \nu} + h_{ 1}(x,u')=0\, in\,\, \Gamma_{ 1}\times (0, \infty),\\ \frac{ \partial v}{ \partial \nu} + h_{ 2}(x,v')+\sigma (x)u=0 \, in\, \,\Gamma_{ 1}\times (0, \infty), \end{ array} \right. $$ and the respective initial conditions at $t=0$. Here $\Omega$ is a bounded open set of $\mathbb{ R}^n$ with boundary $\Gamma$ constituted by two disjoint parts $\Gamma_{ 0}$ and $\Gamma_{ 1}$ and $\nu(x)$ denotes the exterior unit normal vector at $x\in \Gamma_{ 1}$. The functions $h_{ i}(x,s),\,\, (i=1,2)$ are continuous and strongly monotone in $s\in \mathbb{ R}$. The existence of solutions of the above problem is obtained by applying the Galerkin method with a special basis, the compactness method and a result of approximation of continuous functions by Lipschitz continuous functions due to Strauss. The exponential decay of energy follows by using appropriate Lyapunov functional and the multiplier method.}
Categories: math.AP