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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

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 and 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.

[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 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.

[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.

[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.

[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 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.

[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 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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 and 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 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.

[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 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.

[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.

[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 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.

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 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.

[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.

[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.

[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 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.

[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 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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 and 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 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.

[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 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.

[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.

[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 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.

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