Abstract
In this paper we define multivalued semiprocesses and give theorems providing the existence of global attractors for such systems. This theory generalizes the construction of nonautonomous dynamical systems given by V. V. Chepyzhov and M. I. Vishik to the case where the system is not supposed to have a unique solution for each initial state. Further, we apply these theorems to nonautonomous differential inclusions of reaction–diffusion type.
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References
Aubin, J. P. and Cellina, A.: Differential Inclusions, Springer-Verlag, Berlin, 1984.
Aubin, J. P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser, Boston, 1990.
Babin, A. V.: Attractor of the generalized semigroup generated by an elliptic equation in a cylindrical domain, Russian Acad. Sci. Izv. Math. 44(2) (1995), 207–223.
Ball, J. M.: Continuity properties and global attractors of generalized semiflows and the Navier- Stokes equations, J. Nonlinear Sci. 7 (1997), 475–502.
Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976.
Caraballo, T., Langa, J. A. and Valero J.: Global attractors for multivalued random dynamical systems, Nonlinear Anal. (to appear).
Chepyzhov, V. V. and Vishik, M. I.: Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl. 73 (1994), 279–333.
Chepyzhov, V. V. and Vishik, M. I.: Evolution equations and their trajectory attractors, J. Math. Pures Appl. 76 (1997), 913–964.
Gutman, S.: Existence theorems for nonlinear evolution equations, Nonlinear Anal. 11(10) (1987), 1193–1206.
Hale, J. K.: Asymptotic Behavior of Dissipative Systems, Math. Surveys Monographs 25, Amer. Math. Soc., Providence, 1988.
Kapustian, A. V. and Melnik, V. S.: On global attractors of multivalued semidynamical systems and their approximations, Kibernet. Sistem. Anal. 5 (1998), 102–111.
Kapustian, A. V. and Melnik, V. S.: Attractors of multivalued semidynamical systems and their approximations, Dokl. Akad. Nauk Ukrainy 10 (1998), 21–25.
Kapustian, A. V. and Melnik, V. S.: On approximations and estimates of dimension of global compact attractors of multivalued semiflows, Nauchnye Vedomosti of National Technical University of Ukraine “KPI”, 1999 (to appear).
Kapustian, A. V. and Valero, J.: Attractors of multivalued semiflows generated by differential inclusions and their approximations, Abstr. Appl. Anal. (to appear).
Ladyzhenskaya, O. A.: Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
Melnik, V. S.: Multivalued dynamics of nonlinear infinite-dimensional systems, Preprint No. 94-17, Academy of Science of Ukraine, Institute of Cybernetics, 1994.
Melnik, V. S.: Multivalued semiflows and their attractors, Dokl. Akad. Nauk 343(3) (1995), 302–305.
Melnik, V. S.: Families of m-semiflows and their attractors, Dokl. Akad. Nauk 353(2) (1997), 158–162.
Melnik, V. S., Slastikov, V. V. and Vasilkevich, S. I.: On global attractors of multivalued semiprocesses, Dokl. Akad. Nauk Ukrainy 7 (1999), 12–17.
Melnik, V. S., Slastikov, V. V. and Vasilkevich, S. I.: Multivalued semiprocesses in infinite-dimensional spaces and their attractors, submitted.
Melnik, V. S. and Valero, J.: On attractors of multivalued semidynamical systems in infinite-dimensional spaces, Preprint No. 5, Departamento de Matemáticas, Universidad de Murcia, 1997.
Melnik, V. S. and Valero, J.: On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal. 6 (1998), 83–111.
Papageorgiou, N. S.: Evolution inclusions involving a difference term of subdifferentials and applications, Indian J. Pure Appl. Math. 28(5) (1997), 575–610.
Tolstonogov, A. A.: On solutions of evolution inclusions. I, Sibirsk. Mat. Zh. 33(3) (1992), 161–174 (English translation in Siberian Math. J. 33(3) (1992)).
Valero, J.: Existence and dimension of attractors in dynamical systems generated by evolution inclusions, Ph.D. Thesis, Universidad de Murcia, 1997.
Valero, J.: Attractors of parabolic equations without uniqueness, Preprint No. 15, Departamento de Matemáticas, Universidad de Murcia, 1999.
Valero, J.: Finite and infinite-dimensional attractors for multivalued reaction- diffusion equations, Acta Math. Hungar. 88(3) (2000), 239–258.
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Melnik, V.S., Valero, J. On Global Attractors of Multivalued Semiprocesses and Nonautonomous Evolution Inclusions. Set-Valued Analysis 8, 375–403 (2000). https://doi.org/10.1023/A:1026514727329
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DOI: https://doi.org/10.1023/A:1026514727329