In this paper, we provide a general strategy based on Lyapunov functionals to analyze global asymptotic stability of linear infinite-dimensional systems subject to nonlinear dampings under the assumption that the origin of the system is globally asymptotically stable with a linear damping. To do so, we use the fact that for any linear infinite-dimensional system that is globally exponentially stable, there exists a Lyapunov functional. Then, we derive a Lyapunov functional for the nonlinear system, which is the sum of a Lyapunov functional coming from the linear system and another term that compensates the nonlinearity. Our results are then applied to the linearized Korteweg-de Vries equation and some wave equations.