The wave equation with energy critical sources and nonlinear damping defined on a 3D bounded domain is considered. It is shown that the resulting dynamical system admits a global attractor. Under the additional assumption of strong monotonicity of the damping at the origin, it is shown that the originally unstable quintic wave is uniformly stabilised to a finite dimensional and smooth set. Moreover, the existence of exponential attractor is established. In order to handle \enquote{ energy criticality} of both sources and damping, the methods used depend on enhanced dissipation \cite{ Bociu-lasiecka-jde}, energy { \it identity} for weak solutions \cite{ Koch-lasiecka}, an adaptation of Ball's method \cite{ ball}, and the theory of quasi-stable systems \cite{ chueshov-white}.
Categories: math.DS