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

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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

Abstract
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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 - A, 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. 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. 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. 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. 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. 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, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[18]	R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics,, Second edition, (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. 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. 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. Google Scholar
[22]	Y. J. Wang, On the upper semicontinuity of pullback attractors for multi-valued processes,, Quart. Applied Math., 71 (2013), 369. doi: 10.1090/S0033-569X-2013-01306-1. Google Scholar

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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. 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. 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. 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. 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. 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, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[18]	R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics,, Second edition, (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. 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. 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. Google Scholar
[22]	Y. J. Wang, On the upper semicontinuity of pullback attractors for multi-valued processes,, Quart. Applied Math., 71 (2013), 369. doi: 10.1090/S0033-569X-2013-01306-1. Google Scholar

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