The author considers the problem \[ \begin{cases} u_{tt}+\Delta^2u+au_t|u_t|^{m-2}=bu|u|^{p-2}, &x\in \Omega\;\text{ and } t>0,\\ u(x,t)=\partial_\nu u(x,t)=0, &x\in \partial\Omega\;\text{ and } t\geq 0,\\ u(x,0)=\phi(x), u_t(x,0)=\psi(x), &x\in\Omega,\end{cases}\tag{1} \] where \(\Omega\) is a bounded domain of \({\mathbb{R}}^n\) with smooth boundary \(\partial \Omega\), \(\nu\) is the unit outer normal, \(a\), \(b>0\), \(p\), \(m>2\), \(\phi\in H^2_0(\Omega)\) and \(\psi\in L^2(\Omega)\). The author proves that, under the hypotheses \[ p>2\text{ if } n\leq 4,\qquad {2(n-2)\over n-4}>p>2 \text{ if } n\geq 5, \qquad {2n\over n-4}\geq m \text{ if } n\geq 5, \] problem (1) has a unique weak solution. If \(p>m\) and \[ {1\over 2}\int_\Omega(u_t^2+(\Delta u)^2)(x,0) dx-{b\over p}\int_\Omega |u(x,0)|^p dx<0 \] then the solution of (1) blows-up in finite time, while if \(m\geq p\) the solution of (1) exists globally in time.