Abstract
For the class of nonautonomous evolution inclusions with pointwise pseudomonotone multi-valued mappings the dynamics as t → +∞ of all global weak solutions defined on [0, +∞) is studied. The existence of a compact uniform trajectory attractor is proved. The results obtained allow one to study the dynamics of solutions for new classes of evolution inclusions related to nonlinear mathematical models of controlled geophysical and socioeconomic processes and for fields with interaction functions of pseudomonotone type satisfying the condition of polynomial growth and the standard sign condition.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
That is V is a real reflexive separable Banach space continuously and densely embedded into a real Hilbert space H, H is identified with its topologically conjugated space H ∗, V ∗ is a dual space to V. So, there is a chain of continuous and dense embeddings: \(V \subset H \equiv H^{{\ast}}\subset V ^{{\ast}}\) (see, for example, Gajewski et al. [1, Chap. I]).
References
Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare operatorgleichungen und operatordifferentialgleichungen. Akademie, Berlin (1978)
Migórski, S., Ochal, A.: Optimal control of parabolic hemivariational inequalities. J. Glob. Optim. 17, 285–300 (2000)
Zgurovsky, M.Z., Kasyanov, P.O., Kapustyan, O.V., Valero, J., Zadoianchuk, N.V.: Evolution inclusions and variation Inequalities for Earth data processing III. Springer, Berlin (2012)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhauser, Basel (1985)
Balibrea, F., Caraballo, T., Kloeden, P.E., Valero, J.: Recent developments in dynamical systems: three perspectives. Int. J. Bifurcat. Chaos (2010). doi:10.1142/S0218127410027246
Chepyzhov, V.V., Vishik, M.I.: Evolution equations and their trajectory attractors. J. Math. Pures Appl. 76, 913–964 (1997)
Migórski, S.: Boundary hemivariational inequalities of hyperbolic type and applications. J. Glob. Optim. 31(3), 505–533 (2005)
Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum, Boston (2003)
Zgurovsky, M.Z., Mel’nik, V.S., Kasyanov, P.O.: Evolution Inclusions and Variation Inequalities for Earth Data Processing II. Springer, Berlin (2011)
Kasyanov, P.O.: Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity. Math. Notes 92, 205–218 (2012)
Kasyanov, P.O.: Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity. Cybern. Syst. Anal. 47, 800–811 (2011)
Chepyzhov, V.V., Vishik, M.I.: Trajectory and global attractors for 3D Navier-Stokes system. Mat. Zametki (2002). doi:10.1023/A:1014190629738
Chepyzhov, V.V., Vishik, M.I.: Trajectory attractors for evolution equations. C. R. Acad. Sci. Paris. Ser. I 321, 1309–1314 (1995)
Chepyzhov, V.V., Vishik, M.I.: Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete Contin. Dyn. Syst. 27(4), 1498–1509 (2010)
Sell, G.R.: Global attractors for the three-dimensional Navier-Stokes equations. J. Dyn. Differ. Equ. 8(12), 1–33 (1996)
Hale, J.K.: Asymptotic Behavior of Dissipative Systems. AMS, Providence (1988)
Ladyzhenskaya, O.A.: Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)
Mel’nik, V.S., Valero, J.: On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions. Set-Valued Anal. (2000). doi:10.1023/A:1026514727329
Kapustyan, O.V., Kasyanov, P.O., Valero, J.: Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes equations. J. Math. Anal. Appl. (2011). doi:10.1016/j.jmaa.2010.07.040
Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Series in Mathematical Analysis and Applications, vol. 9. Chapman & Hall/CRC, Boca Raton (2005)
Melnik, V.S., Valero, J.: On attractors of multivalued semi-flows and generalized differential equations. Set Valued Anal. 6(1), 83–111 (1998)
Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. Nauka, Moscow (1989) (in Russian)
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68. Springer, New York (1988)
Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Volume II: Applications. Kluwer, Dordrecht (2000)
Kasyanov, P.O., Toscano, L., Zadoianchuk, N.V.: Regularity of weak solutions and their attractors for a parabolic feedback control problem. Set Valued Var. Anal. (2013). doi:10.1007/s11228-013-0233-8
Zgurovsky, M.Z., Kasyanov, P.O., Zadoianchuk (Zadoyanchuk), N.V.: Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem. Appl. Math. Lett. 25(10), 1569–1574 (2012)
Acknowledgements
We thank Professor David Y. Gao for many years of cooperation.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Zgurovsky, M.Z., Kasyanov, P.O. (2015). Evolution Inclusions in Nonsmooth Systems with Applications for Earth Data Processing. In: Gao, D., Ruan, N., Xing, W. (eds) Advances in Global Optimization. Springer Proceedings in Mathematics & Statistics, vol 95. Springer, Cham. https://doi.org/10.1007/978-3-319-08377-3_28
Download citation
DOI: https://doi.org/10.1007/978-3-319-08377-3_28
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08376-6
Online ISBN: 978-3-319-08377-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)