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October  2014, 34(10): 4343-4370. doi: 10.3934/dcds.2014.34.4343

The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain

Yejuan Wang 1, and  Peter E. Kloeden 2,

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
2. 

Institut für Mathematik, Goethe Universität, D-60054 Frankfurt am Main

Received  May 2012 Revised  October 2012 Published  April 2014

The existence of a uniform attractor in a space of higher regularity is proved for the multi-valued process associated with the nonautonomous reaction-diffusion equation on an unbounded domain with delays for which the uniqueness of solutions need not hold. A new method for checking the asymptotical upper-semicompactness of the solutions is used.
Keywords: uniform attractor, Multi-valued process, asymptotical upper-semicompactness, reaction-diffusion equation with delay..
Mathematics Subject Classification: Primary: 35R10, 35B41; Secondary: 47H2.
Citation: Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343
References:
[1]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201. doi: 10.1023/A:1022902802385.  Google Scholar

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T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. doi: 10.1016/j.jde.2004.04.012.  Google Scholar

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T. Caraballo, P. 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.  Google Scholar

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T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous 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.  Google Scholar

[5]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Global attractors for a non-autonomous integro-differential equation in materials with memory, Nonlinear Anal., 73 (2010), 183-201. doi: 10.1016/j.na.2010.03.012.  Google Scholar

[6]

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

[7]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993.  Google Scholar

[8]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Roy. Soc. London, 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753.  Google Scholar

[9]

P. E. Kloeden and B. Schmalfuss, Asymptotical behaviour of nonautonomous difference inclusions, Syst. Cont. Lett., 33 (1998), 275-280. doi: 10.1016/S0167-6911(97)00107-2.  Google Scholar

[10]

D. S. Li and P. E. Kloeden, On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions, J. Differential Equations, 224 (2006), 1-38. doi: 10.1016/j.jde.2005.07.012.  Google Scholar

[11]

D. S. Li, Y. J. Wang and S. Y. Wang, On the dynamics of non-autonomous general dynamical systems and differential inclusions, Set-Valued Anal., 16 (2008), 651-671. doi: 10.1007/s11228-007-0054-8.  Google Scholar

[12]

S. S. Lu, H. Q. Wu and C. K. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719. doi: 10.3934/dcds.2005.13.701.  Google Scholar

[13]

Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.  Google Scholar

[14]

J. Mallet-Paret and G. R. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations, 125 (1996), 385-440. doi: 10.1006/jdeq.1996.0036.  Google Scholar

[15]

J. Mallet-Paret and G. R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037.  Google Scholar

[16]

P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delay in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006. doi: 10.3934/dcds.2010.26.989.  Google Scholar

[17]

F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbbR^N$ with continuous nonlinearity, Asymptotic Anal., 44 (2005), 111-130.  Google Scholar

[18]

R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics, Second edition, Springer-Verlag, New York, 1997.  Google Scholar

[19]

M. I. Vishik and V. V. Chepyzhov, Trajectory and global attractors of the three-dimensional Navier-Stokes systems, Math. Notes, 71 (2002), 177-193; translated from Mat. Zametki, 71 (2002), 194-213. doi: 10.1023/A:1014190629738.  Google Scholar

[20]

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.  Google Scholar

[21]

Y. J. Wang and S. F. Zhou, Kernel sections of multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains, Quart. Applied Math., 67 (2009), 343-378.  Google Scholar

[22]

Y. J. Wang, On the upper semicontinuity of pullback attractors for multi-valued processes, Quart. Applied Math., 71 (2013), 369-399. doi: 10.1090/S0033-569X-2013-01306-1.  Google Scholar

show all references

References:
[1]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201. doi: 10.1023/A:1022902802385.  Google Scholar

[2]

T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. doi: 10.1016/j.jde.2004.04.012.  Google Scholar

[3]

T. Caraballo, P. 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.  Google Scholar

[4]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous 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.  Google Scholar

[5]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Global attractors for a non-autonomous integro-differential equation in materials with memory, Nonlinear Anal., 73 (2010), 183-201. doi: 10.1016/j.na.2010.03.012.  Google Scholar

[6]

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

[7]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993.  Google Scholar

[8]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Roy. Soc. London, 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753.  Google Scholar

[9]

P. E. Kloeden and B. Schmalfuss, Asymptotical behaviour of nonautonomous difference inclusions, Syst. Cont. Lett., 33 (1998), 275-280. doi: 10.1016/S0167-6911(97)00107-2.  Google Scholar

[10]

D. S. Li and P. E. Kloeden, On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions, J. Differential Equations, 224 (2006), 1-38. doi: 10.1016/j.jde.2005.07.012.  Google Scholar

[11]

D. S. Li, Y. J. Wang and S. Y. Wang, On the dynamics of non-autonomous general dynamical systems and differential inclusions, Set-Valued Anal., 16 (2008), 651-671. doi: 10.1007/s11228-007-0054-8.  Google Scholar

[12]

S. S. Lu, H. Q. Wu and C. K. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719. doi: 10.3934/dcds.2005.13.701.  Google Scholar

[13]

Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.  Google Scholar

[14]

J. Mallet-Paret and G. R. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations, 125 (1996), 385-440. doi: 10.1006/jdeq.1996.0036.  Google Scholar

[15]

J. Mallet-Paret and G. R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037.  Google Scholar

[16]

P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delay in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006. doi: 10.3934/dcds.2010.26.989.  Google Scholar

[17]

F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbbR^N$ with continuous nonlinearity, Asymptotic Anal., 44 (2005), 111-130.  Google Scholar

[18]

R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics, Second edition, Springer-Verlag, New York, 1997.  Google Scholar

[19]

M. I. Vishik and V. V. Chepyzhov, Trajectory and global attractors of the three-dimensional Navier-Stokes systems, Math. Notes, 71 (2002), 177-193; translated from Mat. Zametki, 71 (2002), 194-213. doi: 10.1023/A:1014190629738.  Google Scholar

[20]

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.  Google Scholar

[21]

Y. J. Wang and S. F. Zhou, Kernel sections of multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains, Quart. Applied Math., 67 (2009), 343-378.  Google Scholar

[22]

Y. J. Wang, On the upper semicontinuity of pullback attractors for multi-valued processes, Quart. Applied Math., 71 (2013), 369-399. doi: 10.1090/S0033-569X-2013-01306-1.  Google Scholar

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