# American Institute of Mathematical Sciences

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

 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  January 2020 Early access  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.

Citation: Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [13] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [25] V. S. Melnik, Multi-valued dynamics of nonlinear infinite dimensional systems, Preprint of NAS of Ukraine, Institute of Cybernetics, Kyiv, 94 (1994). [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. [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. [28] A. Savostianov and S. Zelik, Recent progress in attractors for quintic wave equations, Mathemaica Bohemica, 139 (2014), 657-665. [29] A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations, Doctoral dissertation, University of Surrey, 2015. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [13] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [25] V. S. Melnik, Multi-valued dynamics of nonlinear infinite dimensional systems, Preprint of NAS of Ukraine, Institute of Cybernetics, Kyiv, 94 (1994). [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. [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. [28] A. Savostianov and S. Zelik, Recent progress in attractors for quintic wave equations, Mathemaica Bohemica, 139 (2014), 657-665. [29] A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations, Doctoral dissertation, University of Surrey, 2015. [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. [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. [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. [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. [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. [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. [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.
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