Geodesic
Recent progress in attractors for quintic wave equations
Mathematica Bohemica, Tome 139 (2014) no. 4, pp. 657-665

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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.
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.
DOI : 10.21136/MB.2014.144142
Classification : 35B30, 35B40, 35B41, 35B45, 35L30, 35L76, 35R11, 37L30
Keywords: damped wave equation; fractional damping; critical nonlinearity; global attractor; smoothness 
Savostianov, Anton; Zelik, Sergey. Recent progress in attractors for quintic wave equations. Mathematica Bohemica, Tome 139 (2014) no. 4, pp. 657-665. doi: 10.21136/MB.2014.144142
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