We consider abstract nonlinear second-order-in-time systems with damping. The nonlinearity is assumed to satisfy a monotonicity condition as well as certain smoothness conditions. Well posedness of solutions is established and several examples of interest are discussed. A nonlinear variation-of-parameters representation for solutions in terms of an associated linear semigroup is also given.
"Global solvability for damped abstract nonlinear hyperbolic systems." Differential Integral Equations 10 (2) 309 - 332, 1997. https://doi.org/10.57262/die/1367526340