In this paper, we prove the existence of the global attractor for the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity.
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[1] | J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations, 17 (1992), 841-866. doi: 10.1080/03605309208820866. |
[2] | A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992. |
[3] | A. V. Balakrishnan and L. W. Taylor, Distributed parameter nonlinear damping models for flight structures, Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989. |
[4] | J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31. |
[5] | V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\mathbb{R}^3$, Discrete Contin. Dynam. Systems, 7 (2001), 719-735. doi: 10.3934/dcds.2001.7.719. |
[6] | A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310. doi: 10.2140/pjm.2002.207.287. |
[7] | A. N. Carvalho, J. W. Cholewa and T. Dlotko, Damped wave equations with fast growing dissipative nonlinearities, Discrete Contin. Dyn. Syst., 24 (2009), 1147-1165. doi: 10.3934/dcds.2009.24.1147. |
[8] | M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. J. Silva and C. M. Webler, Exponential stability for the wave equation with degenerate nonlocal weak damping, Israel J. Math., 219 (2017), 189-213. doi: 10.1007/s11856-017-1478-y. |
[9] | V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, R.I., 2002. |
[10] | I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106. |
[11] | I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262. doi: 10.1016/j.jde.2011.08.022. |
[12] | I. Chueshov, Dynamics of Quasi-stable Dissipative Systems, Springer, Switzerland, 2015. doi: 10.1007/978-3-319-22903-4. |
[13] | I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951. doi: 10.1081/PDE-120016132. |
[14] | I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc., 2008. doi: 10.1090/memo/0912. |
[15] | I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-Posedness and Long Time Dynamics, Springer Science & Business Media, 2010. doi: 10.1007/978-0-387-87712-9. |
[16] | M. A. J. da Silva and V. Narciso, Attractors and their properties for a class of nonlocal extensible beams, Discrete Contin. Dyn. Syst., 35 (2015), 985-1008. doi: 10.3934/dcds.2015.35.985. |
[17] | M. A. J. da Silva and V. Narciso, Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory, 6 (2017), 437-470. doi: 10.3934/eect.2017023. |
[18] | K. Deimling, Nonlinear Functional Analysis, Springer, Berlin 1985. doi: 10.1007/978-3-662-00547-7. |
[19] | P. Ding, Z. Yang and Y. Li, Global attractor of the Kirchhoff wave models with strong nonlinear damping, Appl. Math. Lett., 76 (2018), 40-45. doi: 10.1016/j.aml.2017.07.008. |
[20] | E. Feireisl, Attractors for wave equations with nonlinear dissipation and critical exponent, C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), 551-555. |
[21] | E. Feireisl, Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems, J. Dynam. Differential Equations, 6 (1994), 23-35. doi: 10.1007/BF02219186. |
[22] | E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447. doi: 10.1006/jdeq.1995.1042. |
[23] | J.-M. Ghidaglia and A. Marzocchi, Longtime behaviour of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal., 22 (1991), 879-895. doi: 10.1137/0522057. |
[24] | J.-M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), 273-319. |
[25] | M. Grasselli and V. Pata, On the damped semilinear wave equation with critical exponent, Discrete Contin. Dyn. Syst., (2003), 351–358. |
[26] | J. K. Hale, Functional Differential Equations, Springer-Verlag, New York, 1971. |
[27] | J. K. Hale, Asymptotic behaviour and dynamics in infinite dimensions, in Nonlinear Differential Equations, Pitman, Boston, MA, 1985, 1–42. |
[28] | J. K. Hale and G. Raugel, Attractors for dissipative evolutionary equations, International Conference on Differential Equations, World Sci. Publ., River Edge, NJ, 1 (1993), 3–22. |
[29] | A. Haraux, Two remarks on hyperbolic dissipative problems, Nonlinear Partial Differential Equations and their Applications, 122 (1985), 161-179. |
[30] | A. Haraux, Semi-linear hyperbolic problems in bounded domains, Math. Rep., 3 (1987), 1-281. |
[31] | M. A. Jorge Silva, V. Narciso and A. Vicente, On a beam model related to flight structures with nonlocal energy damping, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3281-3298. doi: 10.3934/dcdsb.2018320. |
[32] | A. K. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101. doi: 10.1016/j.jmaa.2005.05.031. |
[33] | A. K. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719. doi: 10.1016/j.jde.2006.06.001. |
[34] | O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991. doi: 10.1017/CBO9780511569418. |
[35] | H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential Integral Equations, 10 (1997), 1075-1092. |
[36] | P. D. Lax, Functional Analysis, Wiley-Interscience [John Wiley & Sons], New York, 2002. |
[37] | P. P. D. Lazo, Quasi-linear Wave Equation with Damping and Source Terms, Ph.D thesis, Federal University of Rio de Janeiro, Brazil, 1997. |
[38] | P. P. D. Lazo, Global solutions for a nonlinear wave equation, Appl. Math. Comput., 200 (2008), 596-601. doi: 10.1016/j.amc.2007.11.056. |
[39] | Y. Li and Z. Yang, Optimal attractors of the Kirchhoff wave model with structural nonlinear damping, J. Differential Equations, 268 (2020), 7741-7773. doi: 10.1016/j.jde.2019.11.084. |
[40] | Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. |
[41] | V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. doi: 10.1007/s00220-004-1233-1. |
[42] | V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. doi: 10.1088/0951-7715/19/7/001. |
[43] | I. Perai, Multiplicity of Solutions for the p-Laplacian, 1997. |
[44] | G. Raugel, Une équation des ondes avec amortissement non linéaire dans le cas critique en dimension trois, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992), 177-182. |
[45] | G. Raugel, Global attractors in partial differential equations, Handbook of Dynamical Systems, Vol. 2, North-Holland, 2002,885–982. doi: 10.1016/S1874-575X(02)80038-8. |
[46] | R. E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, AMS, Providence, 1997. doi: 10.1090/surv/049. |
[47] | J. Simon, Régularité de la solution d'une équation non linéaire dans RN, Journées d'Analyse Non Linéaire, Lecture Notes in Math., Springer, Berlin, 665 (1978), 205–227. |
[48] | J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. |
[49] | C. D. Sogge, Lectures on Non-linear Wave Equations, 2$^{nd}$ edition, International Press, Boston, MA, 2008. |
[50] | C. Sun, M. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, J. Differential Equations, 227 (2006), 427-443. doi: 10.1016/j.jde.2005.09.010. |
[51] | R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. |
[52] | Z. Yang, P. Ding and L. Li, Longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlinearity, J. Math. Anal. Appl., 442 (2016), 485-510. doi: 10.1016/j.jmaa.2016.04.079. |
[53] | Z. Yang, Z. Liu and P. Niu, Exponential attractor for the wave equation with structural damping and supercritical exponent, Commun. Contemp. Math., 18 (2016), 1550055, 13 pp. doi: 10.1142/S0219199715500558. |
[54] | C. Zhao, C. Zhao and C. Zhong, The global attractor for a class of extensible beams with nonlocal weak damping, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 935-955. doi: 10.3934/dcdsb.2019197. |
[55] | C. Zhao, C. Zhao and C. Zhong, Asymptotic behaviour of the wave equation with nonlocal weak damping and anti-damping, J. Math. Anal. Appl., 490 (2020), 124186, 16 pp. doi: 10.1016/j.jmaa.2020.124186. |