American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution
  • DCDS-B Home
  • This Issue
  • Next Article
    Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions
July  2017, 22(5): 1779-1800. doi: 10.3934/dcdsb.2017106

Attractors for a random evolution equation with infinite memory: Theoretical results

Tomás Caraballo 1, , María J. Garrido-Atienza 1, , Björn Schmalfuss 2, and  José Valero 3,,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain
2. 

Institut für Mathematik, Institut für Stochastik, Ernst Abbe Platz 2,07737-Jena, Germany
3. 

Universidad Miguel Hernandez de Elche, Centro de Investigación Operativa, Avda. Universidad s/n, 03202-Elche (Alicante), Spain

* Corresponding author

Received  April 2016 Revised  June 2016 Published  March 2017

Fund Project: This work has been partially supported by FEDER and Spanish Ministerio de Economĺa y Competitividad, project MTM2015-63723-P, and by Junta de Andalucĺa under Proyecto de Excelencia P12-FQM-1492.

The long-time behavior of solutions (more precisely, the existence of random pullback attractors) for an integro-differential parabolic equation of diffusion type with memory terms, more particularly with terms containing both finite and infinite delays, as well as some kind of randomness, is analyzed in this paper. We impose general assumptions not ensuring uniqueness of solutions, which implies that the theory of multivalued dynamical system has to be used. Furthermore, the emphasis is put on the existence of random pullback attractors by exploiting the techniques of the theory of multivalued nonautonomous/random dynamical systems.

Keywords: Pullback and random attractor, random dynamical system, random delay equation, infinite delay.
Mathematics Subject Classification: 60H15, 60H25, 35K40, 35K41, 35K5.
Citation: Tomás Caraballo, María J. Garrido-Atienza, Björn Schmalfuss, José Valero. Attractors for a random evolution equation with infinite memory: Theoretical results. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1779-1800. doi: 10.3934/dcdsb.2017106
References:
[1]

L. Arnold, Random Dynamical Systems Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. Google Scholar

[2]

T. Caraballo, I. D. Chueshov and J. Real, Pullback attractors for stochastic heat equations in materials with memory, Discrete Cont. Dyn. Systems Series B, 9 (2008), 525-539.   Google Scholar

[3]

T. Caraballo, M. J. Garrido-Atienza and B. Schmalfuß, Existence of exponentially attracting stationary solutions for delay evolution equations, Discrete Contin. Dyn. Syst., 18 (2007), 271-293.  doi: 10.3934/dcds.2007.18.271.  Google Scholar

[4]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.  Google Scholar

[5]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Global attractor for a non-autonomous integro-differential equation in materials with memory, Nonlinear Analysis, 73 (2010), 183-201.  doi: 10.1016/j.na.2010.03.012.  Google Scholar

[6]

T. Caraballo, P. E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl Math Optim, 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.  Google Scholar

[7]

T. Caraballo, J. A. Langa and J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal., 48 (2002), 805-829.  doi: 10.1016/S0362-546X(00)00216-9.  Google Scholar

[8]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, SpringerVerlag, Berlin, 1977. Google Scholar

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[10]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[11]

M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity SIAM Studies in Applied Mathematics 12, SIAM, Philadelphia, 1992. Google Scholar

[12] H. Gajewsky, K. Gröger and K. Zacharias, Nichlineare operatorgleichungen und operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.   Google Scholar
[13]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991. Google Scholar

[14]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. Ⅰ, volume 419 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1997. Google Scholar

[15]

item {ReHrNo87} (MR919738) M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman, Harlow; John Willey, New York, 1987. Google Scholar

[16] J. C. Robinson, Infinite-dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2002.  doi: 10.1007/978-94-010-0732-0.  Google Scholar
[17]

B. Schmalfuß, Attractors for the non-autonomous dynamical systems, In International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), pp. 684{689, World Sci. Publishing, River Edge, NJ, 2000. Google Scholar

[18]

R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1979. Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. Google Scholar

[2]

T. Caraballo, I. D. Chueshov and J. Real, Pullback attractors for stochastic heat equations in materials with memory, Discrete Cont. Dyn. Systems Series B, 9 (2008), 525-539.   Google Scholar

[3]

T. Caraballo, M. J. Garrido-Atienza and B. Schmalfuß, Existence of exponentially attracting stationary solutions for delay evolution equations, Discrete Contin. Dyn. Syst., 18 (2007), 271-293.  doi: 10.3934/dcds.2007.18.271.  Google Scholar

[4]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.  Google Scholar

[5]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Global attractor for a non-autonomous integro-differential equation in materials with memory, Nonlinear Analysis, 73 (2010), 183-201.  doi: 10.1016/j.na.2010.03.012.  Google Scholar

[6]

T. Caraballo, P. E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl Math Optim, 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.  Google Scholar

[7]

T. Caraballo, J. A. Langa and J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal., 48 (2002), 805-829.  doi: 10.1016/S0362-546X(00)00216-9.  Google Scholar

[8]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, SpringerVerlag, Berlin, 1977. Google Scholar

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[10]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[11]

M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity SIAM Studies in Applied Mathematics 12, SIAM, Philadelphia, 1992. Google Scholar

[12] H. Gajewsky, K. Gröger and K. Zacharias, Nichlineare operatorgleichungen und operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.   Google Scholar
[13]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991. Google Scholar

[14]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. Ⅰ, volume 419 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1997. Google Scholar

[15]

item {ReHrNo87} (MR919738) M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Longman, Harlow; John Willey, New York, 1987. Google Scholar

[16] J. C. Robinson, Infinite-dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2002.  doi: 10.1007/978-94-010-0732-0.  Google Scholar
[17]

B. Schmalfuß, Attractors for the non-autonomous dynamical systems, In International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), pp. 684{689, World Sci. Publishing, River Edge, NJ, 2000. Google Scholar

[18]

R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1979. Google Scholar

[1]

Mustapha Yebdri. Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021036
[2]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[3]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319
[4]

Scott Schmieding, Rodrigo Treviño. Random substitution tilings and deviation phenomena. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021020
[5]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074
[6]

Sana Netchaoui, Mohamed Ali Hammami, Tomás Caraballo. Pullback exponential attractors for differential equations with delay. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1345-1358. doi: 10.3934/dcdss.2020367
[7]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352
[8]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284
[9]

Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329
[10]

Lopo F. de Jesus, César M. Silva, Helder Vilarinho. Random perturbations of an eco-epidemiological model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021040
[11]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270
[12]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147
[13]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121
[14]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390
[15]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324
[16]

Josselin Garnier, Knut Sølna. Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1171-1195. doi: 10.3934/dcdsb.2020158
[17]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
[18]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080
[19]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168
[20]

Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 693-716. doi: 10.3934/dcdsb.2020189

2019 Impact Factor: 1.27

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences