Abstract
We apply the theory of multivalued semiflows to a nonlinear parabolic equation of the “reaction–diffusion” type in the case where it is impossible to prove the uniqueness of its solution. A multivalued semiflow is generated by solutions satisfying a certain estimate global in time. We establish the existence of a global compact attractor in the phase space for the multivalued semiflow generated by a nonlinear parabolic equation. We prove that this attractor is an upper-semicontinuous function of a parameter.
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REFERENCES
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations [in Russian], Nauka, Moscow (1989).
J.-L. Lions, Quelques Méthodes de Resolution des Problèmes aux Limites Non Linéaires, Dunod, Paris (1969).
V. S. Melnik and J. Valero, "On attractors of multivalued semi-flows and differential inclusions," Set-Valued Analysis, 6, 83-111 (1998).
O. V. Kapustyan, "An attractor of a semiflow generated by a system of phase-field equations without the uniqueness of a solution," Ukr. Mat. Zh., 51, No. 7, 23-30 (1999).
V. S. Mel'nik, "Multivalued semiflows and their attractors," Dokl. Ross. Akad. Nauk., 337, No. 4, 876-881 (1994).
J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York (1984).
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Kapustyan, O.V., Shkundin, D.V. Global Attractor of One Nonlinear Parabolic Equation. Ukrainian Mathematical Journal 55, 535–547 (2003). https://doi.org/10.1023/B:UKMA.0000010155.48722.f2
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DOI: https://doi.org/10.1023/B:UKMA.0000010155.48722.f2