# American Institute of Mathematical Sciences

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, 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 [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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##### 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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