Energy convergence results for strongly damped nonlinear wave equations (Q1082543)

Let A, B be (unbounded) positive self-adjoint operators on a Hilbert space H. Consider the strongly-damped nonlinear wave equation (SDNWE) \[ u_{tt}+(\alpha A^ 2+\beta)u_ t+B^ 2u=F(u) \] where \(\alpha\), \(\beta\) are nonnegative constants and F is a nonlinear map. We develop criteria under which solutions of the SDNWE converge in a suitable energy norm to a solution of the undamped equation \((\alpha =\beta =0)\) on closed time intervals [0,T]. In our application, the vibrating beam equation, T can be chosen arbitrarily large.