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

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.
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

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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. 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

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. 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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