Asymptotic Translation Uniform Integrability and Multivalued Dynamics of Solutions for Non-autonomous Reaction-Diffusion Equations

  • Michael Z. Zgurovsky
  • Pavlo O. Kasyanov
  • Nataliia V. Gorban
  • Liliia S. Paliichuk
Part of the Understanding Complex Systems book series (UCS)


In this note we introduce asymptotic translation uniform integrability condition for a function acting from a positive semi-axes of time-line to a Banach space. We prove that this condition is equivalent to uniform integrability condition. As a result, we obtain the corollaries for the multivalued dynamics (as time t → +) of solutions for non-autonomous reaction-diffusion equations.


Multivalued Dynamics Uniform Integrability Condition Chepyzhov Weak Solution Case Generation Problem 
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  1. 1.
    Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations (in Russian). Nauka, Moscow (1989)zbMATHGoogle Scholar
  2. 2.
    Balibrea, F., Caraballo, T., Kloeden, P.E., Valero, J.: Recent developments in dynamical systems: three perspectives. Int. J. Bifurcation Chaos (2010).
  3. 3.
    Chepyzhov, V.V., Vishik, M.I.: Evolution equations and their trajectory attractors. J. Math. Pures Appl. 76, 913–964 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics. American Mathematical Society, Providence (2002)zbMATHGoogle Scholar
  5. 5.
    Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare operatorgleichungen und operatordifferentialgleichungen. Akademie-Verlag, Berlin (1978)zbMATHGoogle Scholar
  6. 6.
    Gluzman, M.O., Gorban, N.V., Kasyanov, P.O.: Lyapunov type functions for classes of autonomous parabolic feedback control problems and applications. Appl. Math. Lett. (2014).
  7. 7.
    Gorban, N.V., Kasyanov, P.O.: On regularity of all weak solutions and their attractors for reaction-diffusion inclusion in unbounded domain. Contin. Distrib. Syst. Theory Appl. Solid Mech. Appl. 211, (2014).$_$15
  8. 8.
    Gorban, N.V., Kapustyan, O.V., Kasyanov, P.O.: Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Caratheodory’s nonlinearity. Nonlinear Anal. Theory Methods Appl. 98, 13–26 (2014). MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gorban, N.V., Kapustyan, O.V., Kasyanov, P.O., Paliichuk, L.S.: On global attractors for autonomous damped wave equation with discontinuous nonlinearity. Contin. Distrib. Syst. Theory Appl. Solid Mech. Appl. 211 (2014).$_$16
  10. 10.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68. Springer, New York (1988)Google Scholar
  11. 11.
    Zgurovsky, M.Z., Mel’nik, V.S., Kasyanov, P.O.: Evolution Inclusions and Variation Inequalities for Earth Data Processing II. Springer, Berlin (2011)zbMATHGoogle Scholar
  12. 12.
    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)CrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Michael Z. Zgurovsky
    • 1
  • Pavlo O. Kasyanov
    • 2
  • Nataliia V. Gorban
    • 2
  • Liliia S. Paliichuk
    • 2
  1. 1.National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”KyivUkraine
  2. 2.Institute for Applied System AnalysisNational Technical University of Ukraine, Igor Sikorsky Kyiv Politechnic InstituteKyivUkraine

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