MATHEMATICA BOHEMICA, Vol. 139, No. 4, pp. 657-665, 2014

# Recent progress in attractors for quintic wave equations

## Anton Savostianov, Sergey Zelik

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

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