×
Kasyanov, Pavlo O.; Toscano, Luisa; Zadoianchuk, Nina V.

Long-time behaviour of solutions for autonomous evolution hemivariational inequality with multidimensional “reaction-displacement” law. (English) Zbl 1237.49015

Abstr. Appl. Anal. 2012, Article ID 450984, 21 p. (2012).
Summary: We consider autonomous evolution inclusions and hemivariational inequalities with nonsmooth dependence between determinative parameters of a problem. The dynamics of all weak solutions defined on the positive semiaxis of time is studied. We prove the existence of trajectory and global attractors and investigate their structure. New properties of complete trajectories are justified. We study classes of mathematical models for geophysical processes and fields containing the multidimensional “reaction-displacement” law as one of possible application. The pointwise behavior of such problem solutions on attractor is described.

Cited in 13 Documents

MSC:

49J40 Variational inequalities
Cite Review PDF
Full Text: DOI
Creative Commons CC license icon Creative Commons BY license icon

References:

[1] S. Migórski, “Dynamic hemivariational inequalities in contact mechanics,” Nonlinear Analysis: Theory, Methods and Applications, vol. 63, no. 5-7, pp. e77-e86, 2005. · Zbl 1153.49306 · doi:10.1016/j.na.2005.01.012
[2] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, vol. 188 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1995. · Zbl 0968.49008
[3] P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser, Boston, Mass, USA, 1985. · Zbl 0579.73014
[4] I. D. Chueshov, “Global attractors for non-linear problems of mathematical physics,” Russian Mathematical Surveys, vol. 48, no. 3, pp. 135-162, 1993. · Zbl 0805.58042 · doi:10.1070/RM1993v048n03ABEH001031
[5] F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1983. · Zbl 0582.49001
[6] Yu. A. Dubinskii, “High order nonlinear parabolic equations,” Journal of Soviet Mathematics, vol. 56, no. 4, pp. 2557-2607, 1991. · Zbl 0734.35044 · doi:10.1007/BF01097353
[7] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Travaux et Recherches Mathématiques, No. 21, Dunod, Paris, France, 1972. · Zbl 0298.73001
[8] H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Mathematische Lehrbücher und Monographien, II. Abteilung, Mathematische Monographien, Band 38, Akademie, Berlin, Germany, 1974. · Zbl 0289.47029
[9] D. Goeleven, M. Miettinen, and P. D. Panagiotopoulos, “Dynamic hemivariational inequalities and their applications,” Journal of Optimization Theory and Applications, vol. 103, no. 3, pp. 567-601, 1999. · Zbl 0973.90060 · doi:10.1023/A:1021783924105
[10] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I: Theory, vol. 419 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. · Zbl 0887.47001
[11] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. II: Applications, vol. 500 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. · Zbl 0943.47037
[12] P. Kalita, “Decay of energy for second-order boundary hemivariational inequalities with coercive damping,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 4, pp. 1164-1181, 2011. · Zbl 1215.35111 · doi:10.1016/j.na.2010.09.053
[13] P. O. Kasyanov, V. S. Melnik, and S. Toscano, “Solutions of Cauchy and periodic problems for evolution inclusions with multi-valued w\lambda 0-pseudomonotone maps,” Journal of Differential Equations, vol. 249, no. 6, pp. 1258-1287, 2010. · Zbl 1216.47093 · doi:10.1016/j.jde.2010.05.008
[14] P. O. Kasyanov, “Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity,” Cybernetics and Systems Analysis, vol. 47, no. 5, pp. 800-811, 2011. · Zbl 1300.47084 · doi:10.1007/s10559-011-9359-6
[15] K. Kuttler, “Non-degenerate implicit evolution inclusions,” Electronic Journal of Differential Equations, no. 34, pp. 1-20, 2000. · Zbl 0958.34047
[16] O. A. Ladyzhenskaya, Mathematical Problems of the Dynamics of Viscous Incompressible Fluids, Fizmatgiz, Moscow, Russia, 1961. · Zbl 0131.09402
[17] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, France, 1969. · Zbl 0189.40603
[18] Z. Liu and S. Migórski, “Noncoercive damping in dynamic hemivariational inequality with application to problem of piezoelectricity,” Discrete and Continuous Dynamical Systems B, vol. 9, no. 1, pp. 129-143, 2008. · Zbl 1153.35363
[19] S. Migórski, “Boundary hemivariational inequalities of hyperbolic type and applications,” Journal of Global Optimization, vol. 31, no. 3, pp. 505-533, 2005. · Zbl 1089.49014 · doi:10.1007/s10898-004-7021-9
[20] S. Migórski, “Evolution hemivariational inequalities with applications,” in Handbook of Nonconvex Analysis and Applications, D. Y. Gao and D. Motreanu, Eds., chapter 8, pp. 409-473, International Press, Somerville, Mass, USA, 2010. · Zbl 1225.49016
[21] S. Migorski, “Existence of solutions to nonlinear second order evolution inclusions without and with impulses,” Dynamics of Continuous, Discrete & Impulsive Systems B, vol. 18, no. 4, pp. 493-520, 2011. · Zbl 1237.34115
[22] R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, Cambridge University Press, Cambridge, UK, 2001. · Zbl 0993.76002
[23] S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Some relatively new techniques for nonlinear problems,” Mathematical Problems in Engineering, vol. 2009, Article ID 234849, 25 pages, 2009. · Zbl 1184.35280 · doi:10.1155/2009/234849
[24] M. Aslam Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199-277, 2004. · Zbl 1134.49304 · doi:10.1016/S0096-3003(03)00558-7
[25] M. A. Noor, “On iterative methods for solving a system of mixed variational inequalities,” Applicable Analysis, vol. 87, no. 1, pp. 99-108, 2008. · Zbl 1354.49015 · doi:10.1080/00036810701799777
[26] M. A. Noor, “On a system of general mixed variational inequalities,” Optimization Letters, vol. 3, no. 3, pp. 437-451, 2009. · Zbl 1171.58308 · doi:10.1007/s11590-009-0123-z
[27] M. A. Noor, “On an implicit method for nonconvex variational inequalities,” Journal of Optimization Theory and Applications, vol. 147, no. 2, pp. 411-417, 2010. · Zbl 1202.90253 · doi:10.1007/s10957-010-9717-y
[28] M. A. Noor, “Extended general variational inequalities,” Applied Mathematics Letters, vol. 22, no. 2, pp. 182-186, 2009. · Zbl 1163.49303 · doi:10.1016/j.aml.2008.03.007
[29] M. A. Noor, K. I. Noor, and Z. Huang, “Bifunction hemivariational inequalities,” Journal of Applied Mathematics and Computing, vol. 35, no. 1-2, pp. 595-605, 2011. · Zbl 1211.49013 · doi:10.1007/s12190-010-0380-0
[30] M. A. Noor, K. I. Noor, and E. Al-Said, “Iterative methods for solving nonconvex equilibrium variational inequalities,” Applied Mathematics & Information Sciences, vol. 6, no. 1, pp. 65-69, 2012. · Zbl 1244.65084
[31] M. A. Noor, “Some aspects of extended general variational inequalities,” Abstract and Applied Analysis. In press. · Zbl 1242.49017
[32] P. D. Panagiotopoulos, “Hemivariational inequalities and fan-variational inequalities. New applications and results,” Atti del Seminario Matematico e Fisico dell’Università di Modena, vol. 43, no. 1, pp. 159-191, 1995. · Zbl 0843.49006
[33] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, vol. 143 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2002. · Zbl 1254.37002
[34] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1988. · Zbl 0662.35001
[35] A. A. Tolstonogov, “Solutions of Evolution Inclusions. I,” Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 33, no. 3, pp. 161-174, 1992. · Zbl 0787.34052 · doi:10.1007/BF00970899
[36] A. A. Tolstonogov and Ya. I. Umanskiĭ, “Solutions of evolution inclusions. II,” Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 33, no. 4, pp. 163-174, 1992. · Zbl 0791.34016 · doi:10.1007/BF00971135
[37] N. V. Zadoyanchuk and P. O. Kasyanov, “The Faedo-Galerkin method for second-order evolution inclusions with W\lambda -pseudomonotone mappings,” Ukrainian Mathematical Journal, vol. 61, no. 2, pp. 195-213, 2009. · Zbl 1224.34200 · doi:10.1007/s11253-009-0207-z
[38] N. V. Zadoyanchuk and P. O. Kasyanov, “Analysis and control of second-order differential-operator inclusions with +-coercive damping,” Cybernetics and Systems Analysis, vol. 46, no. 2, pp. 305-313, 2010. · Zbl 1226.47066 · doi:10.1007/s10559-010-9208-z
[39] M. Z. Zgurovsky, V. S. Mel’nik, and P. O. Kasyanov, Evolution Inclusions and Variation Inequalities for Earth Data Processing I. Operator Inclusions and Variation Inequalities for Earth Data Processing, vol. 24 of Advances in Mechanics and Mathematics, Springer, Berlin, Germany, 2011. · Zbl 1213.47001
[40] M. Z. Zgurovsky, V. S. Mel’nik, and P. O. Kasyanov, Evolution Inclusions and Variation Inequalities for Earth Data Processing II. Differential-Operator Inclusions and Evolution Variation Inequalities for Earth Data Processing, vol. 25 of Advances in Mechanics and Mathematics, Springer, Berlin, Germany, 2011. · Zbl 1213.47002
[41] J. M. Ball, “Global attractors for damped semilinear wave equations,” Discrete and Continuous Dynamical Systems A, vol. 10, no. 1-2, pp. 31-52, 2004. · Zbl 1056.37084 · doi:10.3934/dcds.2004.10.31
[42] M. I. Vishik and V. V. Chepyzhov, “Trajectory and global attractors of the three-dimensional Navier-Stokes system,” Matematicheskie Zametki, vol. 71, no. 2, pp. 194-213, 2002. · Zbl 1130.37404 · doi:10.1023/A:1014190629738
[43] J. M. Ball, “Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations,” Journal of Nonlinear Science, vol. 7, no. 5, pp. 475-502, 1997. · Zbl 0903.58020 · doi:10.1007/s003329900037
[44] V. S. Melnik and J. Valero, “On attractors of multivalued semi-flows and differential inclusions,” Set-Valued Analysis, vol. 6, no. 1, pp. 83-111, 1998. · Zbl 0915.58063 · doi:10.1023/A:1008608431399
[45] O. V. Kapustyan and J. Valero, “Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 20, no. 9, pp. 2723-2734, 2010. · Zbl 1202.37023 · doi:10.1142/S0218127410027313
[46] V. V. Chepyzhov and M. I. Vishik, “Evolution equations and their trajectory attractors,” Journal de Mathématiques Pures et Appliquées. Neuvième Série, vol. 76, no. 10, pp. 913-964, 1997. · Zbl 0896.35032 · doi:10.1016/S0021-7824(97)89978-3
[47] J. Valero and A. Kapustyan, “On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 614-633, 2006. · Zbl 1169.35345 · doi:10.1016/j.jmaa.2005.10.042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.