Abstract
Beginning from the pioneering works [3, 52], the theory of global attractors of infinite-dimensional dynamical systems has become one of the main objects for investigation. Since then, deep results about existence, properties, structure, and dimension of global attractors for a wide class of dissipative systems have been obtained (see, e.g., [7, 38, 54, 75, 78]). For the application of this classical theory to partial and functional differential equations, it was necessary to have global existence and uniqueness of solutions of the Cauchy problem for all initial data of the phase space.
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Arrieta JM, Rodríguez-Bernal A, Valero J (2006) Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity. Internat J Bifur Chaos doi:10.1142/S0218127406016586
Aubin J-P, Cellina A (1984) Differential inclusions. Springer, Berlin
Aubin J-P, Frankowska H (1990) Set-valued analysis. Birkhauser, Boston
Babin AV (1995) Attractor of the generalized semigroup generated by an elliptic equation in a cylindrical domain. Russian Academy of Sciences. Izvestiya Math. doi:10.1070/ IM1995v044n02ABEH001594
Babin AV, Vishik MI (1985) Maximal attractors of semigroups corresponding to evolutionary differential equations. Math Sbornik 126:397–419
Babin AV, Vishik MI (1986) Maximal attractors of semigroups, corresponding to evolution differential equations. Math USSR Sb doi:10.1070/SM1986v054n02ABEH002976
Babin AV, Vishik MI (1989) Attractors of evolution equations. Nauka, Moscow
Balibrea F, Caraballo T, Kloeden PE, Valero J (2010) Recent developments in dynamical systems: Three perspectives. Int J Bifur Chaos doi:10.1142/S0218127410027246
Ball JM (1997) Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations. J Nonlinear Sci. doi:10.1007/s003329900037
Ball JM (2004) Global attractors for damped semilinear wave equations. DCDS. doi:10.3934/dcds.2004.10.31
Barbashin EA (1948) On the theoy of generalized dynamical systems. Moskov Gos Ped Inst Uchen Zap 135:110–133
Barbu V (1976) Nonlinear semigroups and differential equations in Banach spaces. Editura Academiei Noordhoff International Publishing, Bucuresti
Bessaih H, Flandoli F (2000) Weak attractor for a dissipative Euler equation. J Dyn Differ Equat. doi:10.1023/A:1009042520953
Birnir B, Svanstedt N (2004) Existence and strong attractors for the Rayleigh-Bénard problem with a large aspect radio. Discrete Contin Dyn Syst. doi:10.3934/dcds.2004.10.53
Borisovich AV, Gelman BI, Myskis AD, Obukhovskii VV (1986) Introduction to the theory of multivalued maps. VGU, Voronezh
Brezis H (1972) Problemes unilateraux. J Math Pures Appl 51:377–406
Brezis H (1984) Análisis funcional. Alianza Editorial, Madrid
Caraballo T, Langa J, Valero J (2002) Global attractors of multivalued random dynamical systems. Nonlinear Anal doi:10.1016/S0362-546X(00)00216-9
Caraballo T, Marin-Rubio P, Robinson JC (2003) A comparison between two theories for multi-valued semiflows and their asymptotic behavior. Set Valued Anal. doi:10.1023/A:1024422619616
Caraballo T, Marín-Rubio P, Valero J (2005) Autonomous and non-autonomous attractors for differential equations with delays. J Differ Equat doi:10.1016/j.jde.2003.09.008
Cheban D, Fakeeh D (1994) Global attractors of the dynamical systems without uniqueness. Sigma, Kishinev
Chepyzhov VV, Vishik MI (1996) Trajectory attractors for reaction-diffusion systems. Topol Meth Nonlinear Anal 7:49–76
Chepyzhov VV, Vishik MI (1997) Evolution equations and their trajectory attractors. J Math Pure Appl. doi:10.1016/S0021-7824(97)89978-3
Chepyzhov VV, Vishik MI (1997) Trajectory attractors for evolution equations. C R Acad Sci Paris 321:1309–1314
Chepyzhov VV, Vishik MI (2002) Attractors for equations of mathematical physics. American Mathematical Society, RI
Chepyzhov VV, Vishik MI (2002) Trajectory and global attractors for 3D Navier–Stokes system. Mat Zametki doi:10.1023/A:1014190629738
Cheskidov A, Foias C (2006) On global attractors of the 3D Navier–Stokes equations. J Differ Equat. doi:10.1016/j.jde.2006.08.021
Chueshov ID (1993) Global attractors of nonlinear problems of the Mathematical Physics. Russ Math Surv. doi:10.1070/RM1993v048n03ABEH001033
Constantin P (2007) On the Euler equations of incompressible fluids. Bull Am Math Soc 44:603–621
Cutland NJ (2005) Global attractors for small samples and germs of 3D Navier–Stokes equations. Nonlinear Anal. doi:10.1016/j.na.2005.02.114
Díaz JI, Hernández J, Tello L (1997) On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in Climatology. J Math Anal Appl. doi:10.1006/jmaa.1997.5691
Elmounir O, Simonolar F (2000) Abstracteurs compacts pur des problèmes d’evolution sans unicité. Annales de la Facult é des Sciences de Toulouse Série 6 IX:631–654
Fedorchuk VV, Filippov VV (1988) General topology. MGU, Moscow
Flandoli F, Schmalfuss B (1999) Weak solutions and attractors for three-dimensional Navier–Stokes equations with nonregular force. J Dyn Differ Equat doi:10.1023/A:1021937715194
Foias C, Temam R (1987) The connection between the Navier–Stokes equations, dynamical systems and turbulence theory. In: Directions in partial differential equations. Academic, New York, pp. 55–73
Gobbino M, Sardella M (1997) On the connectedness of attractors for dynamical systems. J Differ Equat. doi:10.1006/jdeq.1996.3166
Hale JK (1977) Introduction to functional differential equations. Springer, New York
Hale JK (1988) Asymptotic behavior of dissipative systems. AMS, RI
Hale JK, Lasalle JP (1972) Theory of a general class of dissipative processes. J Math Anal Appl. doi:10.1016/0022-247X(72)90233-8
Haraux A (1988) Attractors of asymptotically compact processes and applications to nonlinear partial differential equations. Comm Part Differ Equat 13:1383–1414
Hetzer G (2001) The shift-semiflow of a multivalued equation from climate modeling. Nonlinear Anal doi:10.1016/S0362-546X(01)00412-6
Hu S, Papageorgiou NS (1997) Handbook of multivalued analysis, volume I: Theory. Kluwer, Dordrecht
Iovane G, Kapustyan OV (2006) Global attractors for non-autonomous wave equation without uniqueness of solution. Syst Res Inform Technol 2:107–120
Kapustyan AV, Melnik VS (1999) On global attractors of multivalued semidynamical systems and their approximations. Dokl Akad Nauk 366:445–448
Kapustyan AV, Valero J (2000) Attractors of multivalued semiflows generated by differential inclusions and their approximations. Abstr Appl Anal 5:33–46
Kapustyan AV, Valero J (2006) On the connectedness and asymptotic behavior of solutions of reaction-diffusion systems. J Math Anal Appl doi:10.1016/j.jmaa.2005.10.042
Kapustyan AV, Melnik VS, Valero J (2003) Attractors of multivalued dynamical processes generated by phase-field equations. Int J Biff Chaos doi:10.1142/S0218127403007801
Kapustyan OV, Valero J (2007) Weak and strong attractors for 3D Navier–Stokes system. J Differ Equat. doi:10.1016/j.jde.2007.06.008
Kapustyan OV, Valero J (2009) On the Kneser property for the Ginzburg–Landau equation and the Lotka–Volterra system with diffusion. J Math Anal Appl. doi:10.1016/j.jmaa.2009.04.010
Kapustyan OV, Valero J (2010) Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions. Int J Bifur Chaos. doi:10.1142/S0218127410027313
Kapustyan OV, Mel’nik VS, Valero J, Yasinsky VV (2008) Global atractors of multi-valued dynamical systems and evolution equations without uniqueness. Naukova Dumka, Kyiv
Ladyzhenskaya OA (1972) Dynamical system, generated by Navier–Stokes equations. Zap Nauch Sem LOMI 27:91–115
Ladyzhenskaya OA (1990) Some comments to my papers on the theory of attractors for abstract semigroups. Zap Nauchn Sem LOMI 188:102–112
Ladyzhenskaya OA (1991) Attractors for semigroups and evolution equations. University Press, Cambridge
Malek J, Necas J (1996) A finite-dimensional attractor for three dimensional flow of incompresible fluids. J Differ Equat doi:10.1006/jdeq.1996.0080
Malek J, Prazak D (2002) Large time behavior via the method of l-trajectories. J Differ Equat doi:10.1006/jdeq.2001.4087
Melnik VS (1994) Multivalued dynamics of non-linear infinite-dimensional systems. Preprint 94-71, Acad Sci Ukraine
Melnik VS (1995) Families of multi-valued semiflows and their attractors. Dokl Ross Akad Nauk 343:302–305
Melnik VS (1995) Multivalued semidynamic systems and their attractors. Dokl Akad Nauk Ukraini 2:22–27
Melnik VS, Valero J (1996) On multivalued dynamical systems which are deliveried by evolution inclusions. In: Proceedings of the nonlinear oscillations conference, Praga, 9–13 Sept 1996, pp. 139–142
Melnik VS, Valero J (1997) On attractors of multivalued semidynamical systems generated by evolution inclusions. Dokl Ross Akad Nauk
Melnik VS, Valero J (1997) On attractors of multivalued semidynamical systems in infinite dimensional spaces. Preprint, Universidad de Murcia 5:45
Melnik VS, Valero J (1998) On attractors of multivalued semiflows and differential inclusions. Set Valued Anal. doi:10.1023/A:1008608431399
Mielke A, Zelik SV (2002) Infinite-dimensional trajectory attractors of elliptoc boundary-value problems in cylindrical domains. Russ Math Surv doi:10.1070/RM2002v057n04ABEH000550
Morillas F, Valero J (2005) Attractors for reaction-diffusion equations in R N with continuous nonlinearity. Asymptot Anal 44:111–130
Morillas F, Valero J (2009) A Peano’s theorem and attractors for lattice dynamical systems. Int J Bifur Chaos. doi:10.1142/S0218127409023196
Norman DE (2001) Chemically reacting fluid flows: Weak solutions and global attractors. J Differ Equat doi:10.1006/jdeq.1998.3500
Otani M (1977) On existence of strong solutions for \(\frac{du} {dt} + \partial {\psi }^{1}\left (u\left (t\right )\right ) - \partial {\psi }^{2}\left (u\left (t\right )\right ) \ni f\left (t\right )\). J Fac Sci Univ Tokio Sect IA Math 24:575–605
Otani M (1984) Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, periodic problems. J Differ Equat. doi:10.1016/0022-0396(84)90161-X
Papageorgiuou NS, Papalini F (1996) On the structure of the solution set of evolution inclusions with time-dependent subdifferentials. Acta Math Univ Comenianae 65:33–51
Rossi R, Segatti A, Stefanelli U (2008) Attractors for gradient flows of non convex functionals and appplications. Arch Rational Mech Anal. doi:10.1007/s00205-007-0078-0
Schimperna G (2007) Global attractors for Cahn–Hilliard equations with nonconstant mobility. Nonlinearity. doi:10.1088/0951-7715/20/8/010
Segatti A (2007) On the hyperbolic relaxation of the Cahn–Hilliard equation in 3D: Approximation and long time behaviour. Math Models Meth Appl Sci. doi:10.1142/S0218202507001978
Sell GR (1996) Global attractors for the three-dimensional Navier–Stokes equations. J Dyn Differ Equat. doi:10.1007/BF02218613
Sell GR, You Y (1995) Dynamics of evolutionary equations. Springer, New-York
Shnirelman A (1997) On the nonuniqueness of weak solution of the Euler equation. Comm Pure Appl Math L:261–1286
Shuhong F (1995) Global attractor for general nonautonomous dynamical systems. Nonlinear World 2:191–216
Temam R (1988) Infinite-dimensional dynamical systems in mechanics and physics. Springer, New York
Tolstonogov AA (1992) On solutions of evolution inclusions I. Siberian Math J. doi:10.1007/BF00970899
Tolstonogov AA, Umanskii YaI (1992) On solutions of evolution inclusions II. Siberian Math J. doi:10.1007/BF00971135
Valero J (1994) Attractors of both semidynamic systems and evolutionary inclusions. Dokl Akad Nauk Ukraini 5:7–11
Valero J (1995) On attractors of evolutionary inclusions in Banach spaces. Ukrainian Math J. doi:10.1007/BF01059043
Valero J (2000) Finite and infinite dimensional attractors of multivalued reaction-diffusion equations. Acta Math Hung. doi:10.1023/A:1006769315268
Valero J (2001) Attractors of parabolic equations without uniqueness. J Dyn Differ Equat. doi:10.1023/A:1016642525800
Vrabie II (1997) Compactness methods for nonlinear equations. Pitman Longman, London
Willard S (1970) General topology. Addison-Wesley, MA
Yamazaki N (2004) Attractors of asymptotically periodic multivalued dynamical systems governed by time-dependent subdifferentials. Elec J Differ Equat 107:1–22
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Zgurovsky, M.Z., Kasyanov, P.O., Kapustyan, O.V., Valero, J., Zadoianchuk, N.V. (2012). Abstract Theory of Multivalued Semiflows. In: Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Advances in Mechanics and Mathematics, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28512-7_1
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