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

Previous Article
Asymptotic behaviour of the solutions to a virus dynamics model with diffusion
DCDS-B Home
This Issue
Next Article
A unifying approach to discrete single-species populations models

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

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

Rodrigo Samprogna ^1, and Tomás Caraballo ^2,

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 March 2018 Early access 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.

Keywords: Pullback attractors, unbounded delays, multivalued process, dynamical boundary, p-Laplacian, asymptotic behavior of solutions.

Mathematics Subject Classification: 35B41, 35K55, 37B55.

Citation: Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete and 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.

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.

[1]	Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595
[2]	Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[3]	Linfang Liu, Xianlong Fu. Existence and upper semicontinuity of (L², L^q) pullback attractors for a stochastic p-laplacian equation. Communications on Pure and Applied Analysis, 2017, 6 (2) : 443-474. doi: 10.3934/cpaa.2017023
[4]	Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure and Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475
[5]	Leszek Gasiński. Positive solutions for resonant boundary value problems with the scalar p-Laplacian and nonsmooth potential. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 143-158. doi: 10.3934/dcds.2007.17.143
[6]	Eun Kyoung Lee, R. Shivaji, Inbo Sim, Byungjae Son. Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1139-1154. doi: 10.3934/cpaa.2019055
[7]	Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure and Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371
[8]	Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063
[9]	Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033
[10]	Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922
[11]	Shanming Ji, Yutian Li, Rui Huang, Xuejing Yin. Singular periodic solutions for the p-laplacian ina punctured domain. Communications on Pure and Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019
[12]	Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063
[13]	Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069
[14]	Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593
[15]	Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2117-2138. doi: 10.3934/cpaa.2021060
[16]	Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1121-1147. doi: 10.3934/dcdsb.2021083
[17]	Jacson Simsen, José Valero. Global attractors for $p$-Laplacian differential inclusions in unbounded domains. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3239-3267. doi: 10.3934/dcdsb.2016096
[18]	Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040
[19]	Dimitri Mugnai, Kanishka Perera, Edoardo Proietti Lippi. A priori estimates for the Fractional p-Laplacian with nonlocal Neumann boundary conditions and applications. Communications on Pure and Applied Analysis, 2022, 21 (1) : 275-292. doi: 10.3934/cpaa.2021177
[20]	Anhui Gu. Weak pullback mean random attractors for non-autonomous $ p $-Laplacian equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3863-3878. doi: 10.3934/dcdsb.2020266

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Rodrigo Samprogna Tomás Caraballo
on Google Scholar
Rodrigo Samprogna Tomás Caraballo

2021 Impact Factor: 1.497

Tools

Article outline

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Rodrigo Samprogna Tomás Caraballo
on Google Scholar
Rodrigo Samprogna Tomás Caraballo

[Back to Top]