Analysis of a non-linear beam equation

In this paper we investigate the existence, uniqueness of global solutions and asymptotic behavior for the initial boundary value problem for a non-linear beam equation\begin{equation}\label{eq.1}\left\{\begin{array}{l}u''-\Delta u+\Delta^2u+ |u|^\rho=0\mbox{ in }\Omega\times(0,\infty)\\ u=0,\;\; \displaystyle\frac{\partial u}{\partial\nu}=0 \mbox{ on }\Gamma\times(0,\infty)\\ u(x,0)=u_0(x),\;\;u'(x,0)=u_1(x), x\in\Omega,\end{array}\right.\end{equation}where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $\rho>0$ is a real number, $\nu(x)$ is the exterior unit normal vector at $x\in\Gamma$. MSC Classification: 35K60 , 35F30