# American Institute of Mathematical Sciences

• Previous Article
Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions
• DCDS-B Home
• This Issue
• Next Article
Error in approximation of Lyapunov exponents on inertial manifolds: The Kuramoto-Sivashinsky equation
May  2008, 9(3&4, May): 581-593. doi: 10.3934/dcdsb.2008.9.581

## Equi-Attraction and the continuous dependence of attractors on time delays

 1 FB Mathematik, Johann Wolfgang Goethe Universität, Postfach 11 19 32, D-60054 Frankfurt a.M. 2 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, Spain

Received  September 2006 Revised  January 2007 Published  February 2008

Under appropriate regularity conditions it is shown that the continuous dependence of the global attractors $\mathcal{A}_\tau$ of semi dynamical systems $S^{(\tau)}(t)$ in $C([-\tau,0];Z)$ with $Z$ a Banach space and time delay $\tau \in [T_*,T^$*$]$, where $T_* > 0$, is equivalent to the equi-attraction of the attractors. Examples and counter examples posed in this right framework are provided.
Citation: P.E. Kloeden, Pedro Marín-Rubio. Equi-Attraction and the continuous dependence of attractors on time delays. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 581-593. doi: 10.3934/dcdsb.2008.9.581
 [1] Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, José A. Langa. Equi-attraction and continuity of attractors for skew-product semiflows. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 2949-2967. doi: 10.3934/dcdsb.2016081 [2] Michael Dellnitz, Mirko Hessel-Von Molo, Adrian Ziessler. On the computation of attractors for delay differential equations. Journal of Computational Dynamics, 2016, 3 (1) : 93-112. doi: 10.3934/jcd.2016005 [3] Piotr Kalita, Grzegorz Łukaszewicz, Jakub Siemianowski. On relation between attractors for single and multivalued semiflows for a certain class of PDEs. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1199-1227. doi: 10.3934/dcdsb.2019012 [4] Sana Netchaoui, Mohamed Ali Hammami, Tomás Caraballo. Pullback exponential attractors for differential equations with delay. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1345-1358. doi: 10.3934/dcdss.2020367 [5] Ting Li. Pullback attractors for asymptotically upper semicompact non-autonomous multi-valued semiflows. Communications on Pure and Applied Analysis, 2007, 6 (1) : 279-285. doi: 10.3934/cpaa.2007.6.279 [6] Tomás Caraballo, Francisco Morillas, José Valero. On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 51-77. doi: 10.3934/dcds.2014.34.51 [7] Pengyu Chen, Xuping Zhang. Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4325-4357. doi: 10.3934/dcdsb.2020290 [8] Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291 [9] Julia García-Luengo, Pedro Marín-Rubio, José Real. Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 181-201. doi: 10.3934/dcds.2014.34.181 [10] Tomás Caraballo, P.E. Kloeden, Pedro Marín-Rubio. Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 177-196. doi: 10.3934/dcds.2007.19.177 [11] Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715 [12] Antônio Luiz Pereira, Severino Horácio da Silva. Continuity of global attractors for a class of non local evolution equations. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 1073-1100. doi: 10.3934/dcds.2010.26.1073 [13] Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579 [14] Mohamed Ali Hammami, Lassaad Mchiri, Sana Netchaoui, Stefanie Sonner. Pullback exponential attractors for differential equations with variable delays. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 301-319. doi: 10.3934/dcdsb.2019183 [15] Tomás Caraballo, Gábor Kiss. Attractors for differential equations with multiple variable delays. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1365-1374. doi: 10.3934/dcds.2013.33.1365 [16] Hongyong Cui, Mirelson M. Freitas, José A. Langa. On random cocycle attractors with autonomous attraction universes. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3379-3407. doi: 10.3934/dcdsb.2017142 [17] Mirelson M. Freitas, Anderson J. A. Ramos, Baowei Feng, Mauro L. Santos, Helen C. M. Rodrigues. Existence and continuity of global attractors for ternary mixtures of solids. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3563-3583. doi: 10.3934/dcdsb.2021196 [18] Yangrong Li, Shuang Yang, Guangqing Long. Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021303 [19] Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17 [20] Francisco Balibrea, José Valero. On dimension of attractors of differential inclusions and reaction-diffussion equations. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 515-528. doi: 10.3934/dcds.1999.5.515

2021 Impact Factor: 1.497