Convergence to equilibrium for solutions of an abstract wave equation with general damping function

We prove convergence to a stationary solution as time goes to infinity of solutions to abstract nonlinear wave equation with general damping term and gradient nonlinearity, provided the trajectory is precompact. The energy is supposed to satisfy a Kurdyka–Łojasiewicz gradient inequality. Our aim is to formulate conditions on the function g   as general as possible when the damping is a scalar multiple of the velocity, and this scalar depends on the norm of the velocity, g(|ut|)utg(|ut|)ut. These turn out to be estimates and a coupling condition with the energy but not global monotonicity. When the damping is more general, we need an angle condition.