• Previous Article
    Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients
  • DCDS Home
  • This Issue
  • Next Article
    Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth
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:
[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.

[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.

[3]

A. N. Carvalho, 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.

[4]

A. N. Carvalho, J. W. Cholewa, 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.

[5]

V. V. Chepyzhov, 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.

[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.

[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.

[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.

[9]

I. Chueshov, 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.

[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.

[11]

F. Dell'Oro, V. Pata, Long-term analysis of strongly damped nonlinear wave equations, Nonlinearity, 24 (2011), 3413-3435. doi: 10.1088/0951-7715/24/12/006.

[12]

F. Dell'Oro, V. Pata, Strongly damped wave equations with critical nonlinearities, Nonlinear Analysis, 75 (2012), 5723-5735. doi: 10.1016/j.na.2012.05.019.

[13]

A. E. Green, 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.

[14]

A. E. Green, 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.

[15]

A. E. Green, 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.

[16]

V. Kalantarov, 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.

[17]

V. Kalantarov, A. Savostianov, 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.

[18]

H. Y. Li, 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.

[19]

H. Y. Li, 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.

[20]

M. Nakao, Z. J. Yang, Global attractors for some qusilinear wave equations with a strong dissipation, Adv. Math. Sci. Appl., 17 (2007), 89-105.

[21]

V. Pata, M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. doi: 10.1007/s00220-004-1233-1.

[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.

[23]

A. Savostianov, S. Zelik, Recent progress in attractors for quintic wave equations, Mathemaica Bohemica, 139 (2014), 657-665.

[24]

A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations, Doctoral dissertation, University of Surrey, 2015.

[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.

[26]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[27]

Y. H. Wang, 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.

[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.

[29]

Z. J. Yang, N. Feng, T. F. Ma, Global attracts of the generalized double dispersion, Nonlinear Analysis, 115 (2015), 103-106. doi: 10.1016/j.na.2014.12.006.

[30]

Z. J. Yang, Z. M. Liu, 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.

[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.

[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.

[33]

S. F. Zhou, 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.

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.

[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.

[3]

A. N. Carvalho, 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.

[4]

A. N. Carvalho, J. W. Cholewa, 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.

[5]

V. V. Chepyzhov, 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.

[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.

[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.

[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.

[9]

I. Chueshov, 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.

[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.

[11]

F. Dell'Oro, V. Pata, Long-term analysis of strongly damped nonlinear wave equations, Nonlinearity, 24 (2011), 3413-3435. doi: 10.1088/0951-7715/24/12/006.

[12]

F. Dell'Oro, V. Pata, Strongly damped wave equations with critical nonlinearities, Nonlinear Analysis, 75 (2012), 5723-5735. doi: 10.1016/j.na.2012.05.019.

[13]

A. E. Green, 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.

[14]

A. E. Green, 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.

[15]

A. E. Green, 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.

[16]

V. Kalantarov, 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.

[17]

V. Kalantarov, A. Savostianov, 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.

[18]

H. Y. Li, 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.

[19]

H. Y. Li, 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.

[20]

M. Nakao, Z. J. Yang, Global attractors for some qusilinear wave equations with a strong dissipation, Adv. Math. Sci. Appl., 17 (2007), 89-105.

[21]

V. Pata, M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. doi: 10.1007/s00220-004-1233-1.

[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.

[23]

A. Savostianov, S. Zelik, Recent progress in attractors for quintic wave equations, Mathemaica Bohemica, 139 (2014), 657-665.

[24]

A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations, Doctoral dissertation, University of Surrey, 2015.

[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.

[26]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[27]

Y. H. Wang, 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.

[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.

[29]

Z. J. Yang, N. Feng, T. F. Ma, Global attracts of the generalized double dispersion, Nonlinear Analysis, 115 (2015), 103-106. doi: 10.1016/j.na.2014.12.006.

[30]

Z. J. Yang, Z. M. Liu, 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.

[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.

[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.

[33]

S. F. Zhou, 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.

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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 - A, 2006, 16 (1) : 121-136. doi: 10.3934/dcds.2006.16.121

[6]

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

[7]

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

[8]

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

[9]

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

[10]

Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559

[11]

A. Kh. Khanmamedov. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 119-138. doi: 10.3934/dcds.2011.31.119

[12]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120

[13]

Alexandre Nolasco de Carvalho, Jan W. Cholewa, Tomasz Dlotko. Damped wave equations with fast growing dissipative nonlinearities. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1147-1165. doi: 10.3934/dcds.2009.24.1147

[14]

Hiroshi Takeda. Global existence of solutions for higher order nonlinear damped wave equations. Conference Publications, 2011, 2011 (Special) : 1358-1367. doi: 10.3934/proc.2011.2011.1358

[15]

Takashi Narazaki. Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations. Conference Publications, 2009, 2009 (Special) : 592-601. doi: 10.3934/proc.2009.2009.592

[16]

Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022

[17]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887

[18]

V. Pata, Sergey Zelik. A remark on the damped wave equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 611-616. doi: 10.3934/cpaa.2006.5.611

[19]

Kotaro Tsugawa. Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index. Communications on Pure & Applied Analysis, 2004, 3 (2) : 301-318. doi: 10.3934/cpaa.2004.3.301

[20]

Xiaojie Hou, Yi Li, Kenneth R. Meyer. Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 265-290. doi: 10.3934/dcds.2010.26.265

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (4)
  • HTML views (2)
  • Cited by (0)

Other articles
by authors

[Back to Top]