Periodic solution for a plate operator (Q2518858)

The author investigates the existence and uniqueness of periodic solutions problem for the nonlinear plate problem: \[ w''+\Delta^2w+|w'|^{p-2}w'=f\quad \text{in }Q, \] \[ w=\frac{\partial w}{\partial \nu}=0\quad \text{on }\Sigma,\quad w(0)=w(T),\quad w'(0)=w'(T), \] where \(Q=\Omega\times (0,T)\), \(\Omega\subset \mathbb{R}^2\), \(\Sigma=\Gamma\times (0,T)\), \(\Gamma=\partial\Omega\). A solution \(w\) has the form \(w=u+u_0\), where \(u\) is a weak solution of the problem \[ \frac d{dt}\left(u''+\Delta^2u+|u'|^{p-2}u'\right)=\frac {df}{dt}\quad \text{in }Q, \] \[ u=\frac{\partial u}{\partial \nu}=\quad \text{on }\Sigma,\quad u(0)=u(T),\quad u'(0)=u'(T),\quad \int_0^Tu(s)\,ds=0\;\text{ in }\Omega \] and \(u_0\) solves the Dirichlet-Neumann problem \(\Delta^2u_0=g_0\) in \(\Omega\), \(u_0=\frac{\partial u_0}{\partial \nu}=\;\text{on }\Gamma.\) The main result is the existence and the uniqueness of a periodic solution \(w=u+u_0.\)