Stabilization of the nonlinear damped wave equation via linear weak observability

We consider the problem of energy decay rates for nonlinearly damped abstract infinite dimensional systems. We prove sharp, simple and quasi-optimal energy decay rates through an indirect method, namely a weak observability estimate for the corresponding undamped system. One of the main advantage of these results is that they allow to combine the optimal-weight convexity method of Alabau-Boussouira (Appl Math Optim 51:61–105, 2005) and a methodology of Ammari and Tucsnak (ESAIM COCV 6:361–386, 2001) for weak stabilization by observability. Our results extend to nonlinearly damped systems, those of Ammari and Tucsnak (ESAIM COCV 6:361–386, 2001). At the end, we give an appendix on the weak stabilization of linear evolution systems.