Previous |  Up |  Next

Article

Keywords:
damped wave equation; fractional damping; critical nonlinearity; global attractor; smoothness
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.
References:
[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. MR 1156492 | Zbl 0778.58002
[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
[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. DOI 10.1016/j.anihpc.2008.12.004 | MR 2566711 | Zbl 1198.58012
[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. DOI 10.1090/S0894-0347-08-00596-1 | MR 2393429 | Zbl 1204.35119
[5] Carvalho, A. N., Cholewa, J. W.: Attractors for strongly damped wave equations with critical nonlinearities. Pac. J. Math. 207 (2002), 287-310. DOI 10.2140/pjm.2002.207.287 | MR 1972247 | Zbl 1060.35082
[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. DOI 10.1017/S0004972700040296 | MR 1939206 | Zbl 1020.35059
[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. DOI 10.1016/j.jde.2008.02.011 | MR 2413843 | Zbl 1151.35056
[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
[9] Chen, S., Triggiani, R.: Proof of extensions of two conjectures on structural damping for elastic systems. Pac. J. Math. 136 (1989), 15-55. DOI 10.2140/pjm.1989.136.15 | MR 0971932 | Zbl 0633.47025
[10] Chepyzhov, V. V., Vishik, M. I.: Attractors for Equations of Mathematical Physics. American Mathematical Society Colloquium Publications 49 American Mathematical Society, Providence (2002). MR 1868930 | Zbl 0986.35001
[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. MR 2771816 | Zbl 1216.37026
[12] Chueshov, I., Lasiecka, I.: Von Karman Evolution Equations. Well-posedness and long time dynamics. Springer Monographs in Mathematics Springer, New York (2010). MR 2643040 | Zbl 1298.35001
[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. DOI 10.1017/S0308210500022630 | MR 1361632 | Zbl 0838.35078
[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. DOI 10.1007/s00028-009-0017-7 | MR 2511557 | Zbl 1239.35160
[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. DOI 10.1080/03605300802608247 | MR 2512857 | Zbl 1173.35086
[16] Grillakis, M. G.: Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity. Ann. Math. (2) 132 (1990), 485-509. MR 1078267 | Zbl 0736.35067
[17] Kalantarov, V., Savostianov, A., Zelik, S.: Attractors for damped quintic wave equations in bounded domains. http://arxiv.org/abs/1309.6272
[18] Kalantarov, V., Zelik, S.: Finite-dimensional attractors for the quasi-linear strongly-damped wave equation. J. Differ. Equations 247 (2009), 1120-1155. DOI 10.1016/j.jde.2009.04.010 | MR 2531174 | Zbl 1183.35053
[19] Kapitanski, L.: Minimal compact global attractor for a damped semilinear wave equation. Commun. Partial Differ. Equations 20 (1995), 1303-1323. DOI 10.1080/03605309508821133 | MR 1335752 | Zbl 0829.35014
[20] Kapitanski, L.: Global and unique weak solutions of nonlinear wave equations. Math. Res. Lett. 1 (1994), 211-223. DOI 10.4310/MRL.1994.v1.n2.a9 | MR 1266760 | Zbl 0841.35067
[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. MR 2508165 | Zbl 1221.37158
[22] Moise, I., Rosa, R., Wang, X.: Attractors for non-compact semigroups via energy equations. Nonlinearity 11 (1998), 1369-1393. DOI 10.1088/0951-7715/11/5/012 | MR 1644413 | Zbl 0914.35023
[23] Pata, V., Zelik, S.: A remark on the damped wave equation. Commun. Pure Appl. Anal. 5 (2006), 611-616. DOI 10.3934/cpaa.2006.5.611 | MR 2217604 | Zbl 1140.35533
[24] Pata, V., Zelik, S.: Smooth attractors for strongly damped wave equations. Nonlinearity 19 (2006), 1495-1506. DOI 10.1088/0951-7715/19/7/001 | MR 2229785 | Zbl 1113.35023
[25] Savostianov, A., Zelik, S.: Smooth attractors for the quintic wave equations with fractional damping. Asymptotic Anal. 87 (2014), 191-221. MR 3195728
[26] Shatah, J., Struwe, M.: Regularity results for nonlinear wave equations. Ann. Math. (2) 138 (1993), 503-518. MR 1247991 | Zbl 0836.35096
[27] Zelik, S.: Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete Contin. Dyn. Syst. 11 (2004), 351-392. DOI 10.3934/dcds.2004.11.351 | MR 2083423 | Zbl 1059.35018
Partner of
EuDML logo