Global dynamics of nonlinear wave equations with cubic non-monotone damping (Q558448)

The author deals with the following nonlinear wave equation with a cubic non-monotone damping and associated initial boundary value problem \[ u_{tt}- a\Delta u+ f(u)+ g(u_t)= h(t,x),\quad t>0,\;x\in\Omega, \] \[ u|_{\partial\Omega}= 0,\quad u(0,x)= u_0(x),\quad u_t(0,x)= u_1(x),\quad x\in\Omega, \] where \(\Omega\subset\mathbb{R}^N\), \(N= 1\) or \(2\), is a bounded domain with locally Lipschitz continuous boundary \(\partial\Omega\), and \(g(s)= -\alpha s+\beta s^3\). Under some suitable assumptions on the data, the author shows that the weak solutions exist globally and generate semiflow. Using the asymptotic bootstrap method, the author proves existence of the weak attractor.