We report on new results concerning the global well-posedness, dissipativity and attractors for the quintic wave equations in bounded domains of ${ℝ}^3$ with damping terms of the form ${(-{\Delta }_x)}^{\theta }{\partial }_{t}u$, where $\theta =0$ or $\theta =1/2$. The main ingredient of the work is the hidden extra regularity of solutions that does not follow from energy estimates. Due to the extra regularity of solutions existence of a smooth attractor then follows from the smoothing property when $\theta =1/2$. For $\theta =0$ existence of smooth attractors is more complicated and follows...

In this paper we will give a brief survey of recent regularity results for Fourier integral operators with complex phases. This will include the case of real phase functions. Applications include hyperbolic partial differential equations as well as non-hyperbolic classes of equations. An application to an oblique derivative problem is also given.