Recent progress in attractors for quintic wave equations

Anton Savostianov; Sergey Zelik

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 4, page 657-665
  • ISSN: 0862-7959

Abstract

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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 ℝ 3 with damping terms of the form ( - Δ x ) θ ∂ t u , where θ = 0 or θ = 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 θ = 1 / 2 . For θ = 0 existence of smooth attractors is more complicated and follows from Strichartz type estimates.

How to cite

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Savostianov, Anton, and Zelik, Sergey. "Recent progress in attractors for quintic wave equations." Mathematica Bohemica 139.4 (2014): 657-665. <http://eudml.org/doc/269859>.

@article{Savostianov2014,
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.},
author = {Savostianov, Anton, Zelik, Sergey},
journal = {Mathematica Bohemica},
keywords = {damped wave equation; fractional damping; critical nonlinearity; global attractor; smoothness; damped wave equation; fractional damping; critical nonlinearity; global attractor; smoothness},
language = {eng},
number = {4},
pages = {657-665},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Recent progress in attractors for quintic wave equations},
url = {http://eudml.org/doc/269859},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Savostianov, Anton
AU - Zelik, Sergey
TI - Recent progress in attractors for quintic wave equations
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 4
SP - 657
EP - 665
AB - 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.
LA - eng
KW - damped wave equation; fractional damping; critical nonlinearity; global attractor; smoothness; damped wave equation; fractional damping; critical nonlinearity; global attractor; smoothness
UR - http://eudml.org/doc/269859
ER -

References

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