# American Institute of Mathematical Sciences

2018, 23(2): 509-523. doi: 10.3934/dcdsb.2017195

## Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay

 1 Departamento de Matemática, Centro de Ciências Exatas e de Tecnologia, Universidade Federal de São Carlos, Caixa Postal 676, 13.565-905 São Carlos SP, Brazil 2 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160,41080-Sevilla, Spain

Received  February 2017 Revised  May 2017 Published  December 2017

Fund Project: This research was partially supported by Programa Ciência sem Fronteiras/CNPq 200493/2015- 9, Ministério da Ciência e Tecnologia, Brazil and by the projects MTM2015-63723-P (MINECO, Spain/ FEDER, EU) and P12-FQM-1492 (Junta de Andalucía).

In this work we prove the existence of solution for a p-Laplacian non-autonomous problem with dynamic boundary and infinite delay. We ensure the existence of pullback attractor for the multivalued process associated to the non-autonomous problem we are concerned.

Citation: Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195
##### References:
 [1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [2] T. Caraballo, P. Marín-Rubio, J. Real and J. Valero, Attractors for differential equations with unbounded delays, J. Differential Equations, 239 (2007), 311-342. doi: 10.1016/j.jde.2007.05.015. [3] T. Caraballo, P. Marín-Rubio, J. Real 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. [4] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002. [5] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Communications in Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976. [6] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equations, 2 (2002), 1-19. doi: 10.1007/s00028-002-8077-y. [7] C. Gal and M. Warma, Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions, Diff. and Int. Equations, 23 (2010), 327-358. [8] C. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, Journal of Differential Equations, 253 (2012), 126-166. doi: 10.1016/j.jde.2012.02.010. [9] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. [10] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432. [11] T. Hintermann, Evolution equations with dynamic boundary conditions, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 113 (1989), 43-60. doi: 10.1017/S0308210500023945. [12] F. Li and B. You, Pullback attractors for non-autonomous p-laplacian equations with dynamic flux boundary conditions, Elet. J. of Diff. Equations, 2014 (2014), 1-11. [13] J. L. Lions and E. Megenes, Non-Homogeneous Boundary Value Problems and Applications Vol. Ⅰ, Springer-Verlag Berlin Heidelberg New York, 1972. [14] A. Z. Manitius, Feedback controllers for a wind tunnel model involving a delay: Analytical design and numerical simulation, IEEE Trans. Automat. Control, 29 (1984), 1058-1068. doi: 10.1109/TAC.1984.1103436. [15] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Disc. and Continuous Dynamical Systems Series A, 31 (2011), 779-796. doi: 10.3934/dcds.2011.31.779. [16] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays, Discrete and cont. dynamical systems. Series B, 14 (2010), 655-673. doi: 10.3934/dcdsb.2010.14.655. [17] P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for two-dimensional Navier-Stokes model in an infinite delay case, Nonlinear Analysis, 74 (2011), 2012-2030. doi: 10.1016/j.na.2010.11.008. [18] R. A. Samprogna, K. Schiabel and C. B. Gentile Moussa, Pullback attractors for multivalued process and application to nonautonomous problem with dynamic boundary conditions, Set-Valued and Variational Analysis, accepted, 2017. [19] Y. Wang and P. E. Kloeden, Pullback attractors of a multi-valued process generated by parabolic differential equations with unbounded delays, Nonlinear Analysis, 90 (2013), 86-95. doi: 10.1016/j.na.2013.05.026. [20] L. Yang, M. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasilinear parabolic equations with dynamical boundary conditions, Disc. and Cont. Dynamical Systems B, 17 (2012), 1-11. doi: 10.3934/dcdsb.2012.17.2635. [21] L. Yang, M. Yang and J. Wu, On uniform attractors for non-autonomous p-Laplacian equation with a dynamic boundary condition, Topological Methods in Nonlinear Analysis, 42 (2013), 169-180.

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##### References:
 [1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [2] T. Caraballo, P. Marín-Rubio, J. Real and J. Valero, Attractors for differential equations with unbounded delays, J. Differential Equations, 239 (2007), 311-342. doi: 10.1016/j.jde.2007.05.015. [3] T. Caraballo, P. Marín-Rubio, J. Real 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. [4] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002. [5] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Communications in Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976. [6] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equations, 2 (2002), 1-19. doi: 10.1007/s00028-002-8077-y. [7] C. Gal and M. Warma, Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions, Diff. and Int. Equations, 23 (2010), 327-358. [8] C. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, Journal of Differential Equations, 253 (2012), 126-166. doi: 10.1016/j.jde.2012.02.010. [9] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. [10] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432. [11] T. Hintermann, Evolution equations with dynamic boundary conditions, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 113 (1989), 43-60. doi: 10.1017/S0308210500023945. [12] F. Li and B. You, Pullback attractors for non-autonomous p-laplacian equations with dynamic flux boundary conditions, Elet. J. of Diff. Equations, 2014 (2014), 1-11. [13] J. L. Lions and E. Megenes, Non-Homogeneous Boundary Value Problems and Applications Vol. Ⅰ, Springer-Verlag Berlin Heidelberg New York, 1972. [14] A. Z. Manitius, Feedback controllers for a wind tunnel model involving a delay: Analytical design and numerical simulation, IEEE Trans. Automat. Control, 29 (1984), 1058-1068. doi: 10.1109/TAC.1984.1103436. [15] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Disc. and Continuous Dynamical Systems Series A, 31 (2011), 779-796. doi: 10.3934/dcds.2011.31.779. [16] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays, Discrete and cont. dynamical systems. Series B, 14 (2010), 655-673. doi: 10.3934/dcdsb.2010.14.655. [17] P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for two-dimensional Navier-Stokes model in an infinite delay case, Nonlinear Analysis, 74 (2011), 2012-2030. doi: 10.1016/j.na.2010.11.008. [18] R. A. Samprogna, K. Schiabel and C. B. Gentile Moussa, Pullback attractors for multivalued process and application to nonautonomous problem with dynamic boundary conditions, Set-Valued and Variational Analysis, accepted, 2017. [19] Y. Wang and P. E. Kloeden, Pullback attractors of a multi-valued process generated by parabolic differential equations with unbounded delays, Nonlinear Analysis, 90 (2013), 86-95. doi: 10.1016/j.na.2013.05.026. [20] L. Yang, M. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasilinear parabolic equations with dynamical boundary conditions, Disc. and Cont. Dynamical Systems B, 17 (2012), 1-11. doi: 10.3934/dcdsb.2012.17.2635. [21] L. Yang, M. Yang and J. Wu, On uniform attractors for non-autonomous p-Laplacian equation with a dynamic boundary condition, Topological Methods in Nonlinear Analysis, 42 (2013), 169-180.
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