Large time dynamics in Klein-Gordon equations

The Klein-Gordon equation is a wave equation with an additional mass damping term. In this presentation, I will review some literature about the dynamic of such equation when settled on an unbounded one-dimensional spatial domain. I will further present some new results regarding the global existence and long time behaviour of solutions that are initialy close to constant or periodic equilibria.
Most notably, I will talk about a viscous approximation of this equation, as well as describe how uniformly local orbital stability can be obtained from polar decomposition.
This is joint work with Björn de Rijk, Emile Bukieda and Dorothee Frey.