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July  2017, 22(5): 1779-1800. doi: 10.3934/dcdsb.2017106

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

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.

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. CaraballoI. 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. CaraballoM. 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. CaraballoM. J. Garrido-AtienzaB. 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. CaraballoM. J. Garrido-AtienzaB. 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. CaraballoP. 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. CaraballoJ. 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. GajewskyK. 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. CaraballoI. 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. CaraballoM. 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. CaraballoM. J. Garrido-AtienzaB. 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. CaraballoM. J. Garrido-AtienzaB. 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. CaraballoP. 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. CaraballoJ. 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. GajewskyK. 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

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