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April  2017, 37(4): 2181-2205. doi: 10.3934/dcds.2017094

## Global attractor for a strongly damped wave equation with fully supercritical nonlinearities

 School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001, China

*Corresponding author: Zhijian Yang

Received  July 2016 Revised  October 2016 Published  December 2016

Fund Project: The first author is supported by NSF grant 11271336,11671367

The paper investigates the existence of global attractor for a strongly damped wave equation with fully supercritical nonlinearities: $u_{tt}-Δ u- Δu_t+h(u_t)+g(u)=f(x)$. In the case when the nonlinearities $h(u_t)$ and $g(u)$ are of fully supercritical growth, which leads to that the weak solutions of the equation lose their uniqueness, by introducing the notion of limit solutions and using the theory on the attractor of the generalized semiflow, we establish the existence of global attractor for the subclass of limit solutions of the equation in natural energy space in the sense of strong topology.

Citation: Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094
##### References:
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Dell'Oro, Global attractors for strongly damped wave equations with subcritical-critical nonlinearities, CPAA, 12 (2013), 1015-1027. doi: 10.3934/cpaa.2013.12.1015. Google Scholar [11] F. Dell'Oro and V. Pata, Long-term analysis of strongly damped nonlinear wave equations, Nonlinearity, 24 (2011), 3413-3435. doi: 10.1088/0951-7715/24/12/006. Google Scholar [12] F. Dell'Oro and V. Pata, Strongly damped wave equations with critical nonlinearities, Nonlinear Analysis, 75 (2012), 5723-5735. doi: 10.1016/j.na.2012.05.019. Google Scholar [13] A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. Ⅰ. Classical continuum physics, Proc. R. Soc. Lond. Ser. A, 448 (1995), 335-356. doi: 10.1098/rspa.1995.0020. Google Scholar [14] A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. Ⅱ. Generalized continua, Proc. R. Soc. Lond. Ser. A, 448 (1995), 357-377. doi: 10.1098/rspa.1995.0021. Google Scholar [15] A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. Ⅲ. Mixtures of interacting continua, Proc. R. Soc. Lond. Ser. A, 448 (1995), 379-388. doi: 10.1098/rspa.1995.0022. Google Scholar [16] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. doi: 10.1016/j.jde.2009.04.010. Google Scholar [17] V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré, 17 (2016), 2555-2584. doi: 10.1007/s00023-016-0480-y. Google Scholar [18] H. Y. Li and S. F. Zhou, Global periodic attractor for strongly damped and driven wave equations, Acta Mathematicae Applicatae Sinica, 22 (2006), 75-80. doi: 10.1007/s10255-005-0287-y. Google Scholar [19] H. Y. Li and S. F. Zhou, On non-autonomous strongly damped wave equations with a uniform attractor and some averaging, J. Math. Anal. Appl., 341 (2008), 791-802. doi: 10.1016/j.jmaa.2007.10.051. Google Scholar [20] M. Nakao and Z. J. Yang, Global attractors for some qusilinear wave equations with a strong dissipation, Adv. Math. Sci. Appl., 17 (2007), 89-105. Google Scholar [21] 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. Google Scholar [22] A. Savostianov, Strichartz estimates and smooth attractors for a sub-quintic wave equation with fractional damping in bounded domains, Adv. Differential Equations, 20 (2015), 495-530. Google Scholar [23] A. Savostianov and S. Zelik, Recent progress in attractors for quintic wave equations, Mathemaica Bohemica, 139 (2014), 657-665. Google Scholar [24] A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations, Doctoral dissertation, University of Surrey, 2015.Google Scholar [25] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar [26] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar [27] Y. H. Wang and C. K. Zhong, Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 33 (2013), 3189-3209. doi: 10.3934/dcds.2013.33.3189. Google Scholar [28] Z. J. Yang, Z. M. Liu and P. P. Niu, Exponential attractor for the wave equation with structural damping and supercritical exponent, Communications in Contemporary Mathematics, 18(2016), 1550055, 13pp. doi: 10.1142/S0219199715500558. Google Scholar [29] Z. J. Yang, N. Feng and T. F. Ma, Global attracts of the generalized double dispersion, Nonlinear Analysis, 115 (2015), 103-106. doi: 10.1016/j.na.2014.12.006. Google Scholar [30] Z. J. Yang, Z. M. Liu and N. Feng, Longtime behavior of the semilinear wave equation with gentle dissipation, Discrete Cont. Dyn. Sys. A, 36 (2016), 6557-6580. doi: 10.3934/dcds.2016084. Google Scholar [31] S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Discrete Cont. Dyn. Sys., 11 (2004), 351-392. doi: 10.3934/dcds.2004.11.351. Google Scholar [32] S. F. Zhou, Dimension of the global Attractor for strongly damped nonlinear wave equation, J. Math. Anal. Appl., 233 (1999), 102-115. doi: 10.1006/jmaa.1999.6269. Google Scholar [33] S. F. Zhou and X. M. Fan, Kernel sections for non-autonomous strongly damped wave equations, J. Math. Anal. Appl., 275 (2002), 850-869. doi: 10.1016/S0022-247X(02)00437-7. Google Scholar

show all references

##### References:
 [1] J. M. Ball, Continuity properties and attractors of generalized semiflows and the Navier-Stokes equations, Nonlinear Science, 7 (1997), 475-502. doi: 10.1007/s003329900037. Google Scholar [2] 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. Google Scholar [3] 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. Google Scholar [4] A. N. Carvalho, J. W. Cholewa and T. Dlotko, Damped wave equations with fast growing dissipative nonlinearities, Discrete Contin. Dyn. Syst.:A, 24 (2009), 1147-1165. doi: 10.3934/dcds.2009.24.1147. Google Scholar [5] V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964. doi: 10.1016/S0021-7824(97)89978-3. Google Scholar [6] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, volume 49 of American Mathematical Society Colloquium Publications, (Providence, RI: American Mathematical Society), 2002. Google Scholar [7] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping in Memories of AMS, (Providence, RI: American Mathematical Society), 195 (2008), viii+183 pp. doi: 10.1090/memo/0912. Google Scholar [8] 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. Google Scholar [9] I. Chueshov and A. Rezounenko, Dynamics of second order in time evolution equations with state-dependent delay, Nonlinear Analysis, 123/124 (2015), 126-149. doi: 10.1016/j.na.2015.04.013. Google Scholar [10] F. Dell'Oro, Global attractors for strongly damped wave equations with subcritical-critical nonlinearities, CPAA, 12 (2013), 1015-1027. doi: 10.3934/cpaa.2013.12.1015. Google Scholar [11] F. Dell'Oro and V. Pata, Long-term analysis of strongly damped nonlinear wave equations, Nonlinearity, 24 (2011), 3413-3435. doi: 10.1088/0951-7715/24/12/006. Google Scholar [12] F. Dell'Oro and V. Pata, Strongly damped wave equations with critical nonlinearities, Nonlinear Analysis, 75 (2012), 5723-5735. doi: 10.1016/j.na.2012.05.019. Google Scholar [13] A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. Ⅰ. Classical continuum physics, Proc. R. Soc. Lond. Ser. A, 448 (1995), 335-356. doi: 10.1098/rspa.1995.0020. Google Scholar [14] A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. Ⅱ. Generalized continua, Proc. R. Soc. Lond. Ser. A, 448 (1995), 357-377. doi: 10.1098/rspa.1995.0021. Google Scholar [15] A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. Ⅲ. Mixtures of interacting continua, Proc. R. Soc. Lond. Ser. A, 448 (1995), 379-388. doi: 10.1098/rspa.1995.0022. Google Scholar [16] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155. doi: 10.1016/j.jde.2009.04.010. Google Scholar [17] V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré, 17 (2016), 2555-2584. doi: 10.1007/s00023-016-0480-y. Google Scholar [18] H. Y. Li and S. F. Zhou, Global periodic attractor for strongly damped and driven wave equations, Acta Mathematicae Applicatae Sinica, 22 (2006), 75-80. doi: 10.1007/s10255-005-0287-y. Google Scholar [19] H. Y. Li and S. F. Zhou, On non-autonomous strongly damped wave equations with a uniform attractor and some averaging, J. Math. Anal. Appl., 341 (2008), 791-802. doi: 10.1016/j.jmaa.2007.10.051. Google Scholar [20] M. Nakao and Z. J. Yang, Global attractors for some qusilinear wave equations with a strong dissipation, Adv. Math. Sci. Appl., 17 (2007), 89-105. Google Scholar [21] 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. Google Scholar [22] A. Savostianov, Strichartz estimates and smooth attractors for a sub-quintic wave equation with fractional damping in bounded domains, Adv. Differential Equations, 20 (2015), 495-530. Google Scholar [23] A. Savostianov and S. Zelik, Recent progress in attractors for quintic wave equations, Mathemaica Bohemica, 139 (2014), 657-665. Google Scholar [24] A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations, Doctoral dissertation, University of Surrey, 2015.Google Scholar [25] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar [26] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar [27] Y. H. Wang and C. K. Zhong, Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models, Discrete Contin. Dyn. Syst., 33 (2013), 3189-3209. doi: 10.3934/dcds.2013.33.3189. Google Scholar [28] Z. J. Yang, Z. M. Liu and P. P. Niu, Exponential attractor for the wave equation with structural damping and supercritical exponent, Communications in Contemporary Mathematics, 18(2016), 1550055, 13pp. doi: 10.1142/S0219199715500558. Google Scholar [29] Z. J. Yang, N. Feng and T. F. Ma, Global attracts of the generalized double dispersion, Nonlinear Analysis, 115 (2015), 103-106. doi: 10.1016/j.na.2014.12.006. Google Scholar [30] Z. J. Yang, Z. M. Liu and N. Feng, Longtime behavior of the semilinear wave equation with gentle dissipation, Discrete Cont. Dyn. Sys. A, 36 (2016), 6557-6580. doi: 10.3934/dcds.2016084. Google Scholar [31] S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Discrete Cont. Dyn. Sys., 11 (2004), 351-392. doi: 10.3934/dcds.2004.11.351. Google Scholar [32] S. F. Zhou, Dimension of the global Attractor for strongly damped nonlinear wave equation, J. Math. Anal. Appl., 233 (1999), 102-115. doi: 10.1006/jmaa.1999.6269. Google Scholar [33] S. F. Zhou and X. M. Fan, Kernel sections for non-autonomous strongly damped wave equations, J. Math. Anal. Appl., 275 (2002), 850-869. doi: 10.1016/S0022-247X(02)00437-7. Google Scholar
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