American Institute of Mathematical Sciences

Advanced
January  2020, 25(1): 223-240. doi: 10.3934/dcdsb.2019179

Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness

Zhiming Liu and  Zhijian Yang ,

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

* Corresponding author: Zhijian Yang

Received  December 2018 Revised  March 2019 Published  July 2019

Fund Project: This work is supported by NSFC (Grant No. 11671367).

The paper investigates the existence of global attractors for a few classes of multi-valued operators. We establish some criteria and give their applications to a strongly damped wave equation with fully supercritical nonlinearities and without the uniqueness of solutions. Moreover, the geometrical structure of the global attractors of the corresponding multi-valued operators is shown.

Keywords: Multi-valued operators, generalized semigroup, subclass of weak solutions, global attractor, geometrical structure, strongly damped wave equation.
Mathematics Subject Classification: Primary: 37L05, 37L30; Secondary: 35B33, 35B40, 35B41.
Citation: 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
References:
[1]

A. V. Babin and M. I. Vishik, Maximal attractor of the semigroups corresponding to evolution differential equations, (Russian) Mat. Sb. (N.S.), 126 (1985), 397–419,432.  Google Scholar

[2]

A. V. Babin, Attractor of the generalized semi-group generated by an elliptic equation in a cylindrical domain, Russian Acad. Sci. Izv. Math., 44 (1995), 207-223.  doi: 10.1070/IM1995v044n02ABEH001594.  Google Scholar

[3]

F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, Int. J. Bifurcat. Chaos, 20 (2010), 2591-2636.  doi: 10.1142/S0218127410027246.  Google Scholar

[4]

J. M. Ball, On the asymptotic behavior of generalized processes with applications to nonlinear evolution equations, J. Differential Equations, 27 (1978), 224-265.  doi: 10.1016/0022-0396(78)90032-3.  Google Scholar

[5]

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

[6]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Cont. Dyn. Sys., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[7]

L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860.  doi: 10.3934/dcds.2008.22.835.  Google Scholar

[8]

L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.  doi: 10.1016/j.jde.2010.03.009.  Google Scholar

[9]

T. Caraballo, P. Marín-Rubio and J. C. Robinson, A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Anal., 11 (2003), 297-322.  doi: 10.1023/A:1024422619616.  Google Scholar

[10]

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

[11]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for reaction-diffusion systems, Topological Methods in Nonlinear Analysis, 7 (1996), 49-76.  doi: 10.12775/TMNA.1996.002.  Google Scholar

[12]

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

[13]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[14]

V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088.  doi: 10.3934/dcds.2012.32.2079.  Google Scholar

[15]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of AMS, Amer. Math. Soc. Providence, RI, 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.  Google Scholar

[16]

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

[17]

H. Cui, Y. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Analysis, 128 (2015), 303-324.  doi: 10.1016/j.na.2015.08.009.  Google Scholar

[18]

S. Dashkovskiy, P. Feketa, O. Kapustyan and I. Romaniuk, Invariance and stability of global attractors for multi-valued impulsive dynamical systems, J. Math. Anal. Appl., 458 (2018), 193-218.  doi: 10.1016/j.jmaa.2017.09.001.  Google Scholar

[19]

F. Dell'Oro, Global attractors for strongly damped wave equations with subcritical-critical nonlinearities, Communications on Pure and Applied Analysis, 12 (2013), 1015-1027.  doi: 10.3934/cpaa.2013.12.1015.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

P. Kalita and G. Lukaszewicz, Global attractors for multi-valued semiflows with weak continuity properties, Nonlinear Analysis, 101 (2014), 124-143.  doi: 10.1016/j.na.2014.01.026.  Google Scholar

[24]

A. V. Kapustyan, A. V. Pankov and J. Valero, On global attractors of multi-valued semiflows generated by the 3D Benard system, Set-Valued Var. Anal., 20 (2012), 445-465.  doi: 10.1007/s11228-011-0197-5.  Google Scholar

[25]

V. S. Melnik, Multi-valued dynamics of nonlinear infinite dimensional systems, Preprint of NAS of Ukraine, Institute of Cybernetics, Kyiv, 94 (1994). Google Scholar

[26]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Analysis, 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

E. Vitillaro, On the wave equation with hyperbolic dynamical boundary conditions, interior and boundary damping and supercritical sources, J. Differential Equations, 265 (2018), 4873-4941.  doi: 10.1016/j.jde.2018.06.022.  Google Scholar

[31]

Y. J. Wang and L. Yang, Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness, Discrete Cont. Dyn. Sys. B, 24 (2019), 1961-1987.   Google Scholar

[32]

Z. J. Yang, N. Feng and T. F. Ma, Global attractor for the generalized double dispersion equation, Nonlinear Analysis, 115 (2015), 103-116.  doi: 10.1016/j.na.2014.12.006.  Google Scholar

[33]

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

[34]

Z. J. Yang and Z. M. Liu, Global attractor for a strongly damped wave equation with fully supercritical nonlinearities, Discrete Cont. Dyn. Sys. A, 37 (2017), 2181-2205.  doi: 10.3934/dcds.2017094.  Google Scholar

[35]

M. C. Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM Journal on Mathematical Analysis, 47 (2015), 1530-1561.  doi: 10.1137/140978995.  Google Scholar

[36]

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

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Maximal attractor of the semigroups corresponding to evolution differential equations, (Russian) Mat. Sb. (N.S.), 126 (1985), 397–419,432.  Google Scholar

[2]

A. V. Babin, Attractor of the generalized semi-group generated by an elliptic equation in a cylindrical domain, Russian Acad. Sci. Izv. Math., 44 (1995), 207-223.  doi: 10.1070/IM1995v044n02ABEH001594.  Google Scholar

[3]

F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, Int. J. Bifurcat. Chaos, 20 (2010), 2591-2636.  doi: 10.1142/S0218127410027246.  Google Scholar

[4]

J. M. Ball, On the asymptotic behavior of generalized processes with applications to nonlinear evolution equations, J. Differential Equations, 27 (1978), 224-265.  doi: 10.1016/0022-0396(78)90032-3.  Google Scholar

[5]

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

[6]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Cont. Dyn. Sys., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[7]

L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860.  doi: 10.3934/dcds.2008.22.835.  Google Scholar

[8]

L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.  doi: 10.1016/j.jde.2010.03.009.  Google Scholar

[9]

T. Caraballo, P. Marín-Rubio and J. C. Robinson, A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Anal., 11 (2003), 297-322.  doi: 10.1023/A:1024422619616.  Google Scholar

[10]

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

[11]

V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for reaction-diffusion systems, Topological Methods in Nonlinear Analysis, 7 (1996), 49-76.  doi: 10.12775/TMNA.1996.002.  Google Scholar

[12]

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

[13]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[14]

V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088.  doi: 10.3934/dcds.2012.32.2079.  Google Scholar

[15]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of AMS, Amer. Math. Soc. Providence, RI, 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.  Google Scholar

[16]

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

[17]

H. Cui, Y. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Analysis, 128 (2015), 303-324.  doi: 10.1016/j.na.2015.08.009.  Google Scholar

[18]

S. Dashkovskiy, P. Feketa, O. Kapustyan and I. Romaniuk, Invariance and stability of global attractors for multi-valued impulsive dynamical systems, J. Math. Anal. Appl., 458 (2018), 193-218.  doi: 10.1016/j.jmaa.2017.09.001.  Google Scholar

[19]

F. Dell'Oro, Global attractors for strongly damped wave equations with subcritical-critical nonlinearities, Communications on Pure and Applied Analysis, 12 (2013), 1015-1027.  doi: 10.3934/cpaa.2013.12.1015.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

P. Kalita and G. Lukaszewicz, Global attractors for multi-valued semiflows with weak continuity properties, Nonlinear Analysis, 101 (2014), 124-143.  doi: 10.1016/j.na.2014.01.026.  Google Scholar

[24]

A. V. Kapustyan, A. V. Pankov and J. Valero, On global attractors of multi-valued semiflows generated by the 3D Benard system, Set-Valued Var. Anal., 20 (2012), 445-465.  doi: 10.1007/s11228-011-0197-5.  Google Scholar

[25]

V. S. Melnik, Multi-valued dynamics of nonlinear infinite dimensional systems, Preprint of NAS of Ukraine, Institute of Cybernetics, Kyiv, 94 (1994). Google Scholar

[26]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Analysis, 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

E. Vitillaro, On the wave equation with hyperbolic dynamical boundary conditions, interior and boundary damping and supercritical sources, J. Differential Equations, 265 (2018), 4873-4941.  doi: 10.1016/j.jde.2018.06.022.  Google Scholar

[31]

Y. J. Wang and L. Yang, Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness, Discrete Cont. Dyn. Sys. B, 24 (2019), 1961-1987.   Google Scholar

[32]

Z. J. Yang, N. Feng and T. F. Ma, Global attractor for the generalized double dispersion equation, Nonlinear Analysis, 115 (2015), 103-116.  doi: 10.1016/j.na.2014.12.006.  Google Scholar

[33]

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

[34]

Z. J. Yang and Z. M. Liu, Global attractor for a strongly damped wave equation with fully supercritical nonlinearities, Discrete Cont. Dyn. Sys. A, 37 (2017), 2181-2205.  doi: 10.3934/dcds.2017094.  Google Scholar

[35]

M. C. Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM Journal on Mathematical Analysis, 47 (2015), 1530-1561.  doi: 10.1137/140978995.  Google Scholar

[36]

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

[1]

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
[2]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[3]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015
[4]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270
[5]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[6]

Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033
[7]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[8]

Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Equipartition of energy for nonautonomous damped wave equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 597-613. doi: 10.3934/dcdss.2020364
[9]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020407
[10]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268
[11]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147
[12]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[13]

Liang Huang, Jiao Chen. The boundedness of multi-linear and multi-parameter pseudo-differential operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020291
[14]

Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021025
[15]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171
[16]

Shang Wu, Pengfei Xu, Jianhua Huang, Wei Yan. Ergodicity of stochastic damped Ostrovsky equation driven by white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1615-1626. doi: 10.3934/dcdsb.2020175
[17]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345
[18]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024
[19]

Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020172
[20]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (141)
  • HTML views (322)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences