# American Institute of Mathematical Sciences

November  2016, 21(9): 2949-2967. doi: 10.3934/dcdsb.2016081

## Equi-attraction and continuity of attractors for skew-product semiflows

 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla 2 Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP 3 Instituto de Ciências Matemáticas e de Computacao, Universidade de Săo Paulo-Campus de Săo Carlos, Caixa Postal 668, 13560-970 Săo Carlos SP, Brazil 4 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla

Received  July 2015 Revised  November 2015 Published  October 2016

In this paper we prove the equivalence between equi-attraction and continuity of attractors for skew-product semi-flows, and equi-attraction and continuity of uniform and cocycle attractors associated to non-autonomous dynamical systems. To this aim proper notions of equi-attraction have to be introduced in phase spaces where the driving systems depend on a parameter. Results on the upper and lower-semicontinuity of uniform and cocycle attractors are relatively new in the literature, as a deep understanding of the internal structure of these sets is needed, which is generically difficult to obtain. The notion of lifted invariance for uniform attractors allows us to compare the three types of attractors and introduce a common framework in which to study equi-attraction and continuity of attractors. We also include some results on the rate of attraction to the associated attractors.
Citation: 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
##### References:
 [1] A. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992. [2] M. C. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, J. Differential Equations, 257 (2014), 490-522. doi: 10.1016/j.jde.2014.04.008. [3] T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976. doi: 10.1016/j.na.2009.09.037. [4] T. Caraballo, J. C. Jara, J. A. Langa and Z. Liu, Morse decomposition of attractors for non-autonomous dynamical systems, Adv. Nonlinear Stud., 13 (2013), 309-329. doi: 10.1515/ans-2013-0204. [5] T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513. [6] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. [7] A. N. Carvalho, J. A. Langa and J. C. Robinson, On the continuity of pullback attractors for evolution processes, Nonlinear Anal., 71 (2009), 1812-1824. doi: 10.1016/j.na.2009.01.016. [8] A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: continuity of local stable and unstable manifolds, J. Differential Equations, 233 (2007), 622-653. doi: 10.1016/j.jde.2006.08.009. [9] A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numer. Funct. Anal. Optim., 27 (2006), 785-829. doi: 10.1080/01630560600882723. [10] V. V. Chephyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. [11] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lecture Note Series, 278. Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404. [12] L. Desheng and P. E. Kloeden, Equi-attraction and the continuous dependence of pullback attractors on parameters, Stoch. Dyn., 4 (2004), 373-384. doi: 10.1142/S0219493704001061. [13] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. [14] J. K. Hale, L. T. Magalhães and W. M. Oliva, An Introduction to Infinite-Dimensional Dynamical Systems-Geometric Theory, Applied Mathematical Sciences Vol. 47, Springer-Verlag, 1984. doi: 10.1007/0-387-22896-9_9. [15] J. K. Hale, X. B. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp. 50 (1988), 89-123. doi: 10.1090/S0025-5718-1988-0917820-X. [16] J. K. Hale and G. Raugel, Lower semi-continuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl. 154 (1989), 281-326. doi: 10.1007/BF01790353. [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics Vol. 840, Springer-Verlag, 1981. [18] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112. doi: 10.1142/S0219493703000632. [19] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176. [20] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418. [21] C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, Vol. 2002, Springer, 2010. doi: 10.1007/978-3-642-14258-1. [22] J. C. Robinson, Infinite-dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001 doi: 10.1007/978-94-010-0732-0. [23] R. J. Sacker and G. R. Sell, Skew-product flows, finite extensions of minimal transformation groups and almost periodic differential equations, Bull. Amer. Math. Soc., 79 (1973), 802-805. doi: 10.1090/S0002-9904-1973-13325-7. [24] R. J. Sacker and G. R. Sell, Lifting properties in skew-product flows with applications to differential equations, Mem. Amer. Math. Soc., 11 (1977), iv+67 pp. [25] R. J. Sacker, Skew-product Dynamical Systems, Dynamical systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I., (1974)), Vol. II, 175-179. Academic Press, New York, 1976. [26] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [27] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

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##### References:
 [1] A. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992. [2] M. C. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, J. Differential Equations, 257 (2014), 490-522. doi: 10.1016/j.jde.2014.04.008. [3] T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976. doi: 10.1016/j.na.2009.09.037. [4] T. Caraballo, J. C. Jara, J. A. Langa and Z. Liu, Morse decomposition of attractors for non-autonomous dynamical systems, Adv. Nonlinear Stud., 13 (2013), 309-329. doi: 10.1515/ans-2013-0204. [5] T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513. [6] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. [7] A. N. Carvalho, J. A. Langa and J. C. Robinson, On the continuity of pullback attractors for evolution processes, Nonlinear Anal., 71 (2009), 1812-1824. doi: 10.1016/j.na.2009.01.016. [8] A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: continuity of local stable and unstable manifolds, J. Differential Equations, 233 (2007), 622-653. doi: 10.1016/j.jde.2006.08.009. [9] A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numer. Funct. Anal. Optim., 27 (2006), 785-829. doi: 10.1080/01630560600882723. [10] V. V. Chephyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. [11] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lecture Note Series, 278. Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404. [12] L. Desheng and P. E. Kloeden, Equi-attraction and the continuous dependence of pullback attractors on parameters, Stoch. Dyn., 4 (2004), 373-384. doi: 10.1142/S0219493704001061. [13] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. [14] J. K. Hale, L. T. Magalhães and W. M. Oliva, An Introduction to Infinite-Dimensional Dynamical Systems-Geometric Theory, Applied Mathematical Sciences Vol. 47, Springer-Verlag, 1984. doi: 10.1007/0-387-22896-9_9. [15] J. K. Hale, X. B. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp. 50 (1988), 89-123. doi: 10.1090/S0025-5718-1988-0917820-X. [16] J. K. Hale and G. Raugel, Lower semi-continuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl. 154 (1989), 281-326. doi: 10.1007/BF01790353. [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics Vol. 840, Springer-Verlag, 1981. [18] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112. doi: 10.1142/S0219493703000632. [19] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176. [20] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418. [21] C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, Vol. 2002, Springer, 2010. doi: 10.1007/978-3-642-14258-1. [22] J. C. Robinson, Infinite-dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001 doi: 10.1007/978-94-010-0732-0. [23] R. J. Sacker and G. R. Sell, Skew-product flows, finite extensions of minimal transformation groups and almost periodic differential equations, Bull. Amer. Math. Soc., 79 (1973), 802-805. doi: 10.1090/S0002-9904-1973-13325-7. [24] R. J. Sacker and G. R. Sell, Lifting properties in skew-product flows with applications to differential equations, Mem. Amer. Math. Soc., 11 (1977), iv+67 pp. [25] R. J. Sacker, Skew-product Dynamical Systems, Dynamical systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I., (1974)), Vol. II, 175-179. Academic Press, New York, 1976. [26] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [27] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
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