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

Zhiming Liu; Zhijian Yang

doi:10.3934/dcdsb.2019179

Article Contents

2020, Volume 25, Issue 1: 223-240. Doi: 10.3934/dcdsb.2019179

This issue Previous Article Boundedness and stabilization in a two-species chemotaxis system with two chemicals Next Article Analytical formula and dynamic profile of mRNA distribution

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
^* Corresponding author: Zhijian Yang

Received: November 30, 2018

Revised: February 28, 2019

Early access: July 2019

Published: January 2020

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 37L05, 37L30; Secondary: 35B33, 35B40, 35B41.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.