April  2019, 24(4): 1961-1987. doi: 10.3934/dcdsb.2018257

Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

* Corresponding author

Dedicated to the memory of professor V.S. Mel'nik

Received  May 2017 Revised  May 2018 Published  April 2019 Early access  October 2018

Fund Project: This work was supported by NSF of China (Grants No. 41875084, 11571153), and the Fundamental Research Funds for the Central Universities under Grant Nos. lzujbky-2016-100 and lzujbky-2018- it58.

We first prove the existence of a compact positively invariant set which exponentially attracts any bounded set for abstract multi-valued semidynamical systems. Then, we apply the abstract theory to handle retarded ordinary differential equations and lattice dynamical systems, as well as reactiondiffusion equations with infinite delays. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true.

Citation: Yejuan Wang, Lin Yang. Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1961-1987. doi: 10.3934/dcdsb.2018257
References:
[1]

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.

[2]

A. V. Babin and B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dynam. Differential Equations, 7 (1995), 567-590.  doi: 10.1007/BF02218725.

[3]

T. CaraballoP. 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.

[4]

T. CaraballoP. Marí­n-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41.  doi: 10.1016/j.jde.2003.09.008.

[5]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuß and J. Valero, Nonautonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.

[6]

T. CaraballoF. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.

[7]

C. Cavaterra and M. Grasselli, Robust exponential attractors for population dynamics models with infinite time delay, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1051-1076.  doi: 10.3934/dcdsb.2006.6.1051.

[8]

D. N. Cheban and D. S. Fakeeh, Global attractors of infinite-dimensional dynamical systems â…¢, (Russian), Izv. Akad. Nauk Respub. Moldova Mat., (1995), 3-13,113,115. 

[9]

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

[10]

I. Chueshov and A. Rezounenko, Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay, Commun. Pur. Appl. Anal., 14 (2015), 1685-1704.  doi: 10.3934/cpaa.2015.14.1685.

[11]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985. doi: 10.1007/978-3-662-00547-7.

[12]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley & Sons, New-York, 1994.

[13]

M. Efendiev and A. Miranville, Finite dimensional attractors for a class of reaction-diffusion equations in Rn with a strong nonlinearity, Discrete Contin. Dynam. Systems., 5 (1999), 399-424.  doi: 10.3934/dcds.1999.5.399.

[14]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  doi: 10.1002/mana.200310186.

[15]

S. GattiM. GrasselliV. Pata and M. Squassina, Robust exponential attractors for a family of nonconserved phase-field systems with memory, Discrete Contin. Dyn. Syst., 12 (2005), 1019-1029.  doi: 10.3934/dcds.2005.12.1019.

[16]

M. Grasselli and D. Pražák, Exponential attractors for a class of reaction-diffusion problems with time delays, J. Evol. Equ., 7 (2007), 649-667.  doi: 10.1007/s00028-007-0326-7.

[17]

M. Grasselli and V. Pata, Robust exponential attractors for a phase-field system with memory, J. Evol. Equ., 5 (2005), 465-483.  doi: 10.1007/s00028-005-0199-6.

[18]

J. K. Hale, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer, New York, 1977.

[19]

S. Habibi, Estimates on the dimension of an exponential attractor for a delay differential equation, Math. Slovaca, 64 (2014), 1237-1248.  doi: 10.2478/s12175-014-0272-0.

[20]

O. V. KapustyanP. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term, Discrete Contin. Dyn. Syst., 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.

[21]

P. E. Kloeden, Asymptotic invariance and limit sets of general control systems, J. Differential Equations, 19 (1975), 91-105.  doi: 10.1016/0022-0396(75)90021-2.

[22]

A. V. Kapustyan and V. S. Mel'nik, Global attractors of multivalued semidynamical systems and their approximation, Cybernet. Syst. Anal., 34 (1998), 719-725.  doi: 10.1007/BF02667045.

[23]

A. V. Kapustyan and V. S. Mel'nik, On the global attractors of multivalued semidynamical systems and their approximations, Dokl. Akad. Nauk, 366 (1999), 445-448. 

[24]

Y. J. LiH. Q. Wu and T. G. Zhao, Necessary and sufficient conditions for the existence of exponential attractors for semigroups, and applications, Nonlinear Anal., 75 (2012), 6297-6305.  doi: 10.1016/j.na.2012.07.003.

[25]

V. S. Mel'nik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Values Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.

[26]

F. Morillas and J. Valero, Peano's theorem and attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 19 (2009), 557-578.  doi: 10.1142/S0218127409023196.

[27]

J. C. Oliveira and J. M. Pereira, Global attractor for a class of nonlinear lattices, J. Math. Anal. Appl., 370 (2010), 726-739.  doi: 10.1016/j.jmaa.2010.04.074.

[28]

Y. F. Shao and Y. H. Zhou, Existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays, Nonlinear Anal., 74 (2011), 1107-1118.  doi: 10.1016/j.na.2010.09.042.

[29]

M. Y. Sui and Y. J. Wang, Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations, Appl. Math. Comput., 242 (2014), 315-327.  doi: 10.1016/j.amc.2014.05.045.

[30]

Y. J. Wang and S. F. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differential Equations, 232 (2007), 573-622.  doi: 10.1016/j.jde.2006.07.005.

[31]

Y. J. Wang and M. Y. Sui, Finite lattice approximation of infinite lattice systems with delays and non-Lipschitz nonlinearities, Asymptotic. Anal., 106 (2018), 169-203.  doi: 10.3233/ASY-171444.

[32]

M. Z. ZgurovskyPavlo O. Kasyanov and N. V. Zadoianchuk, Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem, Appl. Math. Lett., 25 (2012), 1569-1574.  doi: 10.1016/j.aml.2012.01.016.

[33]

S. Zelik, The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's A-entropy, Math. Nachr., 232 (2001), 129-179.  doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.0.CO;2-T.

[34]

S. F. Zhou and X. Y. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dyn. Diff. Equat., 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.

[35]

J. ZhangP. E. KloedenM. H. Yang and C. K. Zhong, Global exponential κ-dissipative semigroups and exponential attraction, Discrete Contin. Dyn. Syst., 37 (2017), 3487-3502.  doi: 10.3934/dcds.2017148.

[36]

C. K. Zhong and W. S. Niu, On the Z2-index of the global attractor for a class of p-Laplacian equations, Nonlinear Anal., 73 (2010), 3698-3704.  doi: 10.1016/j.na.2010.07.022.

show all references

Dedicated to the memory of professor V.S. Mel'nik

References:
[1]

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.

[2]

A. V. Babin and B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dynam. Differential Equations, 7 (1995), 567-590.  doi: 10.1007/BF02218725.

[3]

T. CaraballoP. 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.

[4]

T. CaraballoP. Marí­n-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41.  doi: 10.1016/j.jde.2003.09.008.

[5]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuß and J. Valero, Nonautonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.

[6]

T. CaraballoF. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.

[7]

C. Cavaterra and M. Grasselli, Robust exponential attractors for population dynamics models with infinite time delay, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1051-1076.  doi: 10.3934/dcdsb.2006.6.1051.

[8]

D. N. Cheban and D. S. Fakeeh, Global attractors of infinite-dimensional dynamical systems â…¢, (Russian), Izv. Akad. Nauk Respub. Moldova Mat., (1995), 3-13,113,115. 

[9]

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

[10]

I. Chueshov and A. Rezounenko, Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay, Commun. Pur. Appl. Anal., 14 (2015), 1685-1704.  doi: 10.3934/cpaa.2015.14.1685.

[11]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985. doi: 10.1007/978-3-662-00547-7.

[12]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley & Sons, New-York, 1994.

[13]

M. Efendiev and A. Miranville, Finite dimensional attractors for a class of reaction-diffusion equations in Rn with a strong nonlinearity, Discrete Contin. Dynam. Systems., 5 (1999), 399-424.  doi: 10.3934/dcds.1999.5.399.

[14]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  doi: 10.1002/mana.200310186.

[15]

S. GattiM. GrasselliV. Pata and M. Squassina, Robust exponential attractors for a family of nonconserved phase-field systems with memory, Discrete Contin. Dyn. Syst., 12 (2005), 1019-1029.  doi: 10.3934/dcds.2005.12.1019.

[16]

M. Grasselli and D. Pražák, Exponential attractors for a class of reaction-diffusion problems with time delays, J. Evol. Equ., 7 (2007), 649-667.  doi: 10.1007/s00028-007-0326-7.

[17]

M. Grasselli and V. Pata, Robust exponential attractors for a phase-field system with memory, J. Evol. Equ., 5 (2005), 465-483.  doi: 10.1007/s00028-005-0199-6.

[18]

J. K. Hale, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer, New York, 1977.

[19]

S. Habibi, Estimates on the dimension of an exponential attractor for a delay differential equation, Math. Slovaca, 64 (2014), 1237-1248.  doi: 10.2478/s12175-014-0272-0.

[20]

O. V. KapustyanP. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term, Discrete Contin. Dyn. Syst., 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.

[21]

P. E. Kloeden, Asymptotic invariance and limit sets of general control systems, J. Differential Equations, 19 (1975), 91-105.  doi: 10.1016/0022-0396(75)90021-2.

[22]

A. V. Kapustyan and V. S. Mel'nik, Global attractors of multivalued semidynamical systems and their approximation, Cybernet. Syst. Anal., 34 (1998), 719-725.  doi: 10.1007/BF02667045.

[23]

A. V. Kapustyan and V. S. Mel'nik, On the global attractors of multivalued semidynamical systems and their approximations, Dokl. Akad. Nauk, 366 (1999), 445-448. 

[24]

Y. J. LiH. Q. Wu and T. G. Zhao, Necessary and sufficient conditions for the existence of exponential attractors for semigroups, and applications, Nonlinear Anal., 75 (2012), 6297-6305.  doi: 10.1016/j.na.2012.07.003.

[25]

V. S. Mel'nik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Values Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.

[26]

F. Morillas and J. Valero, Peano's theorem and attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 19 (2009), 557-578.  doi: 10.1142/S0218127409023196.

[27]

J. C. Oliveira and J. M. Pereira, Global attractor for a class of nonlinear lattices, J. Math. Anal. Appl., 370 (2010), 726-739.  doi: 10.1016/j.jmaa.2010.04.074.

[28]

Y. F. Shao and Y. H. Zhou, Existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays, Nonlinear Anal., 74 (2011), 1107-1118.  doi: 10.1016/j.na.2010.09.042.

[29]

M. Y. Sui and Y. J. Wang, Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations, Appl. Math. Comput., 242 (2014), 315-327.  doi: 10.1016/j.amc.2014.05.045.

[30]

Y. J. Wang and S. F. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differential Equations, 232 (2007), 573-622.  doi: 10.1016/j.jde.2006.07.005.

[31]

Y. J. Wang and M. Y. Sui, Finite lattice approximation of infinite lattice systems with delays and non-Lipschitz nonlinearities, Asymptotic. Anal., 106 (2018), 169-203.  doi: 10.3233/ASY-171444.

[32]

M. Z. ZgurovskyPavlo O. Kasyanov and N. V. Zadoianchuk, Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem, Appl. Math. Lett., 25 (2012), 1569-1574.  doi: 10.1016/j.aml.2012.01.016.

[33]

S. Zelik, The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's A-entropy, Math. Nachr., 232 (2001), 129-179.  doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.0.CO;2-T.

[34]

S. F. Zhou and X. Y. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dyn. Diff. Equat., 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.

[35]

J. ZhangP. E. KloedenM. H. Yang and C. K. Zhong, Global exponential κ-dissipative semigroups and exponential attraction, Discrete Contin. Dyn. Syst., 37 (2017), 3487-3502.  doi: 10.3934/dcds.2017148.

[36]

C. K. Zhong and W. S. Niu, On the Z2-index of the global attractor for a class of p-Laplacian equations, Nonlinear Anal., 73 (2010), 3698-3704.  doi: 10.1016/j.na.2010.07.022.

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