Anton Savostianov,Sergey Zelik, University of Surrey, Department of Mathematics, Guildford, Surrey GU2 7XH, United Kingdom, e-mail:a.savostianov@surrey.ac.uk,s.zelik@surrey.ac.uk

**Abstract:** We report on new results concerning the global well-posedness, dissipativity and attractors for the quintic wave equations in bounded domains of $\mathbb R^3$ 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 from Strichartz type estimates.

**Keywords:** damped wave equation; fractional damping; critical nonlinearity; global attractor; smoothness

**Classification (MSC 2010):** 35B40, 35B45

**Full text** available as PDF.

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.