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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 |
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.
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.
|
show all references
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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