Exponential attractors for the sup-cubic wave equation with nonlocal damping. (English) Zbl 07850945
Summary: We study the long-time dynamics of a wave equation with nonlocal weak damping, nonlocal weak anti-damping and sup-cubic nonlinearity. Based on the Strichartz estimates in a bounded domain, we obtain the global well-posedness of the Shatah-Struwe solutions. To overcome the difficulties brought by the nonlinear weak damping term, we present a new-type Gronwall’s lemma to obtain the dissipative for the Shatah-Struwe solutions semigroup of this equation. Finally, we establish the existence of a time-dependent exponential attractor with the help of a more general criteria constructed by the quasi-stable technique.
MSC:
| 37L30 | Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems |
| 37L05 | General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations |
| 47H20 | Semigroups of nonlinear operators |
Keywords:
wave equation; nonlocal damping; sup-cubic nonlinearity; Shatah-Struwe solution; exponential attractorReferences:
| [1] | Ball, JM, Global attractors for damped semilinear wave equations, Discr. Cont. Dyn. Syst., 10, 31-52, 2004 · Zbl 1056.37084 · doi:10.3934/dcds.2004.10.31 |
| [2] | Blair, MD; Smith, HF; Sogge, CD, Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H. Poincaré Anal. Non Linaire, 26, 5, 1817-1829, 2009 · Zbl 1198.58012 · doi:10.1016/j.anihpc.2008.12.004 |
| [3] | Brezis, H., Analyse Fonctionnelle. Théorie et Applications, 1983, Paris: Masson, Paris · Zbl 0511.46001 |
| [4] | Burq, N.; Lebeau, G.; Planchon, F., Global existence for energy critical waves in 3-D domains, J. AMS, 21, 3, 831-845, 2008 · Zbl 1204.35119 |
| [5] | Burq, N.; Lebeau, G.; Planchon, F., Global existence for energy critical waves in 3-D domains: Neumann boundary conditions, Am. J. Math., 131, 6, 1715-1742, 2009 · Zbl 1184.35210 · doi:10.1353/ajm.0.0084 |
| [6] | Carvalho, AN; Sonner, S., Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pure. Appl. Anal., 12, 3047-3071, 2013 · Zbl 1390.37126 · doi:10.3934/cpaa.2013.12.3047 |
| [7] | Carvalho, AN; Sonner, S., Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Commun. Pure. Appl. Anal., 13, 1114-1165, 2014 · Zbl 1336.37059 |
| [8] | Cavalcanti, MM; Cavalcanti, VND; Ma, TF, Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains, Differ. Int. Equ., 17, 5-6, 495-510, 2004 · Zbl 1174.74320 |
| [9] | Chang, Q.Q., Li, D.D., Sun, C.Y., Zelik, S.: Deterministic and random attractors for a wave equations with sign changing damping. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 87, 161-210 (2023); translation in Izv. Math. 87, 154-199 (2023) · Zbl 1522.35093 |
| [10] | Cavalcanti, MM; Cavalcanti, VND; Silva, MAJ; Webler, CM, Exponential stability for the wave equation with degenerate nonlocal weak damping, Israel J. Math., 219, 1, 189-213, 2017 · Zbl 1375.35042 · doi:10.1007/s11856-017-1478-y |
| [11] | Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., vol. 49, Amer. Math. Soc., Providence, RI (2002) · Zbl 0986.35001 |
| [12] | Chueshov, I., Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1, 86-106, 2010 · Zbl 1216.37026 |
| [13] | Chueshov, I., Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differ. Equ., 252, 1229-1262, 2012 · Zbl 1237.37053 · doi:10.1016/j.jde.2011.08.022 |
| [14] | Chueshov, I., Dynamics of Quasi-Stable Dissipative Systems, 2015, New York: Springer, New York · Zbl 1362.37001 · doi:10.1007/978-3-319-22903-4 |
| [15] | Chueshov, I.; Lasiecka, I., Attractors for second-order evolution equations with nonlinear damping, J. Dynam. Differ. Equ., 16, 469-512, 2004 · Zbl 1072.37054 · doi:10.1007/s10884-004-4289-x |
| [16] | Chueshov, I., Lasiecka, I.: Long-time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc., vol. 195, book 912, Providence (2008) · Zbl 1151.37059 |
| [17] | Feireisl, E., Global attractors for damped wave equations with supercritical exponent, J. Differ. Equ., 116, 431-447, 1995 · Zbl 0819.35097 · doi:10.1006/jdeq.1995.1042 |
| [18] | Kalantarov, V.; Savostianov, A.; Zelik, S., Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré, 17, 2555-2584, 2016 · Zbl 1356.35055 · doi:10.1007/s00023-016-0480-y |
| [19] | Kapitanski, L., Minimal compact global attractor for a damped semilinear wave equation, Comm. Partial Differ. Equ., 20, 1303-1323, 1995 · Zbl 0829.35014 · doi:10.1080/03605309508821133 |
| [20] | Lange, H.; Perla Menzala, G., Rates of decay of a nonlocal beam equation, Differ. Int. Equ., 10, 6, 1075-1092, 1997 · Zbl 0940.35191 |
| [21] | Lions, JL, Quelques Méthodes de Résolution des Problémes aux Limites Non linéaires, 1969, Paris: Dunod, Paris · Zbl 0189.40603 |
| [22] | Majda, A.: Introduction to PDEs and Waves for the Atmosphere and Oceans, AMS, Providence, RI (2003) · Zbl 1278.76004 |
| [23] | Mei, XY; Sun, CY, Uniform attractors for a weakly damped wave equation with sup-cubic nonlinearity, Appl. Math. Lett., 95, 179-185, 2019 · Zbl 1423.35041 · doi:10.1016/j.aml.2019.04.003 |
| [24] | Mei, XY; Savostianov, A.; Sun, CY; Zelik, S., Infinite energy solutions for weakly damped quintic wave equations in \(\mathbb{R}^3 \), Trans. Amer. Math. Soc., 374, 5, 3093-3129, 2021 · Zbl 1460.35040 · doi:10.1090/tran/8317 |
| [25] | Pata, V.; Zelik, S., Smooth attractors for strongly damped wave equations, Nonlinearity, 19, 1495-1506, 2006 · Zbl 1113.35023 · doi:10.1088/0951-7715/19/7/001 |
| [26] | Pražák, D., On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, J. Dyn. Differ. Equ., 14, 764-776, 2002 · Zbl 1030.35017 · doi:10.1023/A:1020756426088 |
| [27] | Pucci, P.; Saldi, S., Asymptotic stability for nonlinear damped Kirchhoff systems involving the fractional p-Laplacian operator, J. Differ. Equ., 263, 5, 2375-2418, 2017 · Zbl 1368.35034 · doi:10.1016/j.jde.2017.02.039 |
| [28] | Robinson, JC, Infinite-Dimensional Dyanamical Systems, 2001, Cambridge: Cambridge University Press, Cambridge · Zbl 0980.35001 · doi:10.1007/978-94-010-0732-0 |
| [29] | Savostianov, A., Strichartz estimates and smooth attractors for a sub-quintic wave equation with fractional damping in bounded domains, Adv. Differ. Equ., 20, 495-530, 2015 · Zbl 1316.35047 |
| [30] | Savostianov, A.: Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations, (Doctoral dissertation), University of Surrey (2015) |
| [31] | Silva, MAJ; Narciso, V., Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory, 6, 437-470, 2017 · Zbl 1366.35185 · doi:10.3934/eect.2017023 |
| [32] | Simon, J., Compact sets in the space \(L^p(0, T; B)\), Ann. Mat. Pura Appl., 146, 65-96, 1987 · Zbl 0629.46031 · doi:10.1007/BF01762360 |
| [33] | Sogge, C., Lectures on Non-linear Wave Equations, 2008, Boston: International Press, Boston · Zbl 1165.35001 |
| [34] | Strichartz, R., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44, 705-714, 1977 · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1 |
| [35] | Sun, CY; Cao, DM; Duan, JQ, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19, 2645-2665, 2006 · Zbl 1113.35133 · doi:10.1088/0951-7715/19/11/008 |
| [36] | Sun, CY; Cao, DM; Duan, JQ, Uniform attractors for non-autonomous wave equations with nonlinear damping, SIAM J. Appl. Dynam. Syst., 6, 293-318, 2007 · Zbl 1210.35160 · doi:10.1137/060663805 |
| [37] | Sun, CY; Yang, MH; Zhong, CK, Global attractors for the wave equation with nonlinear damping, J. Differ. Equ., 227, 427-443, 2006 · Zbl 1101.35021 · doi:10.1016/j.jde.2005.09.010 |
| [38] | Yang, ZJ; Li, YN, Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous Kirchhoff wave models, Discr. Cont. Dyn. Syst., 38, 2629-2653, 2018 · Zbl 1398.37085 · doi:10.3934/dcds.2018111 |
| [39] | Zelik, SV, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Comm. Pure Appl. Anal., 3, 921-934, 2004 · Zbl 1197.35168 · doi:10.3934/cpaa.2004.3.921 |
| [40] | Zhao, C.Y., Zhao, C.X., Zhong, C.K.: Asymptotic behavior of the wave equation with nonlocal weak damping and anti-damping. J. Math. Anal. Appl. 490 124186-124202 (2020) · Zbl 1441.35066 |
| [41] | Zhao, CY; Zhong, CK; Yan, SL, The global attractor for a class of extensible beams with nonlocal weak damping, Discr. Cont. Dyn. Syst. B, 25, 935-955, 2020 · Zbl 1437.35104 |
| [42] | Zheng, SM, Nonlinear Evolution Equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, 2004, USA: CRC Press, USA · Zbl 1085.47058 |
| [43] | Zhou, F., Li, H.F.: Dynamics of the non-autonomous wave equations with nonlocal weak damping and critical nonlinearity. J. Dyn. Differ. Equ. (Under review) |
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