On a nonlinear degenerate evolution equation with strong damping (Q1196610)

The nonlinear degenerate evolution equation with strong damping \[ K(x,t)u_{tt}-\Delta u-\Delta u_ t+F(u)=0\quad\text{in}\quad Q=\Omega\times]0,T[, \] \[ u(x,0)=u_ 0,\quad (Ku')(x,0)=0\quad\text{in}\quad\Omega,\quad u(x,t)=0\quad\text{on} \quad \Sigma=\Gamma\times]0,T[ \] is considered, where \(K\) is a function with \(K(x,t)\geq 0\), \(K(x,0)=0\) and \(F\) is a continuous real function satisfying \(sF(s)\geq 0\) for all \(s\in\mathbb{R}\), \(\Omega\) is a bounded domain of \(\mathbb{R}^ n\) with smooth boundary \(\Gamma\). The existence of a global weak solution is proved in two parts: \(F\) Lipschitzian and derivable except in a finite number of points, \(F\) continuous.