American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth
  • DCDS Home
  • This Issue
  • Next Article
    Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients
April  2017, 37(4): 2181-2205. doi: 10.3934/dcds.2017094

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

Zhijian Yang , and  Zhiming Liu

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.

Keywords: Strongly damped wave equation, subclass of limit solutions, generalized semiflow, supercritical nonlinearities, global attractor.
Mathematics Subject Classification: Primary:35B40, 35B41; Secondary:35B33, 35B65, 37L30.
Citation: Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094
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

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

[1]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006
[2]

Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179
[3]

Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1015-1027. doi: 10.3934/cpaa.2013.12.1015
[4]

Sergey Zelik. Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 351-392. doi: 10.3934/dcds.2004.11.351
[5]

Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070
[6]

Piotr Kokocki. Homotopy invariants methods in the global dynamics of strongly damped wave equation. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3227-3250. doi: 10.3934/dcds.2016.36.3227
[7]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065
[8]

Brahim Alouini. Global attractor for a one dimensional weakly damped half-wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2655-2670. doi: 10.3934/dcdss.2020410
[9]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[10]

Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305
[11]

Boling Guo, Zhaohui Huo. The global attractor of the damped, forced generalized Korteweg de Vries-Benjamin-Ono equation in $L^2$. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 121-136. doi: 10.3934/dcds.2006.16.121
[12]

Fengjuan Meng, Chengkui Zhong. Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 217-230. doi: 10.3934/dcdsb.2014.19.217
[13]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345
[14]

Pengyan Ding, Zhijian Yang. Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2019, 18 (2) : 825-843. doi: 10.3934/cpaa.2019040
[15]

Fabrizio Colombo, Davide Guidetti. Identification of the memory kernel in the strongly damped wave equation by a flux condition. Communications on Pure & Applied Analysis, 2009, 8 (2) : 601-620. doi: 10.3934/cpaa.2009.8.601
[16]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019
[17]

Fuqin Sun, Mingxin Wang. Non-existence of global solutions for nonlinear strongly damped hyperbolic systems. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 949-958. doi: 10.3934/dcds.2005.12.949
[18]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270
[19]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015
[20]

Milena Stanislavova. On the global attractor for the damped Benjamin-Bona-Mahony equation. Conference Publications, 2005, 2005 (Special) : 824-832. doi: 10.3934/proc.2005.2005.824

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (101)
  • HTML views (69)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy