| Title:
|
Recent progress in attractors for quintic wave equations (English) |
| Author:
|
Savostianov, Anton |
| Author:
|
Zelik, Sergey |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
4 |
| Year:
|
2014 |
| Pages:
|
657-665 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
damped wave equation |
| Keyword:
|
fractional damping |
| Keyword:
|
critical nonlinearity |
| Keyword:
|
global attractor |
| Keyword:
|
smoothness |
| MSC:
|
35B30 |
| MSC:
|
35B40 |
| MSC:
|
35B41 |
| MSC:
|
35B45 |
| MSC:
|
35L30 |
| MSC:
|
35L76 |
| MSC:
|
35R11 |
| MSC:
|
37L30 |
| idZBL:
|
Zbl 06433689 |
| idMR:
|
MR3306855 |
| DOI:
|
10.21136/MB.2014.144142 |
| . |
| Date available:
|
2015-02-04T09:25:54Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144142 |
| . |
| Reference:
|
[1] Babin, A. V., Vishik, M. I.: Attractors of Evolution Equations.Studies in Mathematics and Its Applications 25 North-Holland, Amsterdam (1992), translated and revised from the 1989 Russian original. Zbl 0778.58002, MR 1156492 |
| Reference:
|
[2] Ball, J. M.: Global attractors for damped semilinear wave equations. Partial differential equations and applications.Discrete Contin. Dyn. Syst. 10 (2004), 31-52. MR 2026182 |
| Reference:
|
[3] Blair, M. D., Smith, H. F., Sogge, C. D.: Strichartz estimates for the wave equation on manifolds with boundary.Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009), 1817-1829. Zbl 1198.58012, MR 2566711, 10.1016/j.anihpc.2008.12.004 |
| Reference:
|
[4] Burq, N., Lebeau, G., Planchon, F.: Global existence for energy critical waves in 3-{D} domains.J. Am. Math. Soc. 21 (2008), 831-845. Zbl 1204.35119, MR 2393429, 10.1090/S0894-0347-08-00596-1 |
| Reference:
|
[5] Carvalho, A. N., Cholewa, J. W.: Attractors for strongly damped wave equations with critical nonlinearities.Pac. J. Math. 207 (2002), 287-310. Zbl 1060.35082, MR 1972247, 10.2140/pjm.2002.207.287 |
| Reference:
|
[6] Carvalho, A. N., Cholewa, J. W.: Local well posedness for strongly damped wave equations with critical nonlinearities.Bull. Aust. Math. Soc. 66 (2002), 443-463. Zbl 1020.35059, MR 1939206, 10.1017/S0004972700040296 |
| Reference:
|
[7] Carvalho, A. N., Cholewa, J. W., Dlotko, T.: Strongly damped wave problems: Bootstrapping and regularity of solutions.J. Differ. Equations 244 (2008), 2310-2333. Zbl 1151.35056, MR 2413843, 10.1016/j.jde.2008.02.011 |
| Reference:
|
[8] Chen, S., Triggiani, R.: Gevrey class semigroups arising from elastic systems with gentle dissipation: The case $0<\alpha<1/2$.Proc. Am. Math. Soc. 110 (1990), 401-415. MR 1021208 |
| Reference:
|
[9] Chen, S., Triggiani, R.: Proof of extensions of two conjectures on structural damping for elastic systems.Pac. J. Math. 136 (1989), 15-55. Zbl 0633.47025, MR 0971932, 10.2140/pjm.1989.136.15 |
| Reference:
|
[10] Chepyzhov, V. V., Vishik, M. I.: Attractors for Equations of Mathematical Physics.American Mathematical Society Colloquium Publications 49 American Mathematical Society, Providence (2002). Zbl 0986.35001, MR 1868930 |
| Reference:
|
[11] Chueshov, I.: Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping.J. Abstr. Differ. Equ. Appl. (electronic only) 1 (2010), 86-106. Zbl 1216.37026, MR 2771816 |
| Reference:
|
[12] Chueshov, I., Lasiecka, I.: Von Karman Evolution Equations. Well-posedness and long time dynamics.Springer Monographs in Mathematics Springer, New York (2010). Zbl 1298.35001, MR 2643040 |
| Reference:
|
[13] Feireisl, E.: Asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent.Proc. R. Soc. Edinb., Sect. A, Math. 125 (1995), 1051-1062. Zbl 0838.35078, MR 1361632, 10.1017/S0308210500022630 |
| Reference:
|
[14] Grasselli, M., Schimperna, G., Segatti, A., Zelik, S.: On the 3{D} Cahn-{H}illiard equation with inertial term.J. Evol. Equ. 9 (2009), 371-404. Zbl 1239.35160, MR 2511557, 10.1007/s00028-009-0017-7 |
| Reference:
|
[15] Grasselli, M., Schimperna, G., Zelik, S.: On the 2{D} Cahn-{H}illiard equation with inertial term.Commun. Partial Differ. Equations 34 (2009), 137-170. Zbl 1173.35086, MR 2512857, 10.1080/03605300802608247 |
| Reference:
|
[16] Grillakis, M. G.: Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity.Ann. Math. (2) 132 (1990), 485-509. Zbl 0736.35067, MR 1078267 |
| Reference:
|
[17] Kalantarov, V., Savostianov, A., Zelik, S.: Attractors for damped quintic wave equations in bounded domains.http://arxiv.org/abs/1309.6272. |
| Reference:
|
[18] Kalantarov, V., Zelik, S.: Finite-dimensional attractors for the quasi-linear strongly-damped wave equation.J. Differ. Equations 247 (2009), 1120-1155. Zbl 1183.35053, MR 2531174, 10.1016/j.jde.2009.04.010 |
| Reference:
|
[19] Kapitanski, L.: Minimal compact global attractor for a damped semilinear wave equation.Commun. Partial Differ. Equations 20 (1995), 1303-1323. Zbl 0829.35014, MR 1335752, 10.1080/03605309508821133 |
| Reference:
|
[20] Kapitanski, L.: Global and unique weak solutions of nonlinear wave equations.Math. Res. Lett. 1 (1994), 211-223. Zbl 0841.35067, MR 1266760, 10.4310/MRL.1994.v1.n2.a9 |
| Reference:
|
[21] Miranville, A., Zelik, S.: Attractors for dissipative partial differential equations in bounded and unbounded domains.Handbook of Differential Equations: Evolutionary Equations IV Elsevier/North-Holland, Amsterdam (2008), 103-200 C. M. Dafermos et al. Zbl 1221.37158, MR 2508165 |
| Reference:
|
[22] Moise, I., Rosa, R., Wang, X.: Attractors for non-compact semigroups via energy equations.Nonlinearity 11 (1998), 1369-1393. Zbl 0914.35023, MR 1644413, 10.1088/0951-7715/11/5/012 |
| Reference:
|
[23] Pata, V., Zelik, S.: A remark on the damped wave equation.Commun. Pure Appl. Anal. 5 (2006), 611-616. Zbl 1140.35533, MR 2217604, 10.3934/cpaa.2006.5.611 |
| Reference:
|
[24] Pata, V., Zelik, S.: Smooth attractors for strongly damped wave equations.Nonlinearity 19 (2006), 1495-1506. Zbl 1113.35023, MR 2229785, 10.1088/0951-7715/19/7/001 |
| Reference:
|
[25] Savostianov, A., Zelik, S.: Smooth attractors for the quintic wave equations with fractional damping.Asymptotic Anal. 87 (2014), 191-221. MR 3195728, 10.3233/ASY-131208 |
| Reference:
|
[26] Shatah, J., Struwe, M.: Regularity results for nonlinear wave equations.Ann. Math. (2) 138 (1993), 503-518. Zbl 0836.35096, MR 1247991 |
| Reference:
|
[27] Zelik, S.: Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities.Discrete Contin. Dyn. Syst. 11 (2004), 351-392. Zbl 1059.35018, MR 2083423, 10.3934/dcds.2004.11.351 |
| . |
