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On Regularity of All Weak Solutions and Their Attractors for Reaction-Diffusion Inclusion in Unbounded Domain

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Continuous and Distributed Systems

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 211))

Abstract

We consider the reaction-diffusion equation with multivalued function of interaction in an unbounded domain. Conditions on the parameters of the problem can not guarantee the uniqueness of the solution of the Cauchy problem. In this work we focus on the study of long-term forecasts of the state functions of reaction-diffusion equation with use of the theory of global attractors for multivalued semiflows. It is obtained the results of the existence and properties of all weak solutions. We obtain the standard a priori estimates for weak solutions of the investigated problem, prove the existence of weak solutions, the existence of global and trajectory attractors for the problem in phase and extended phase spaces respectively. We provide the regularity properties for all globally defined weak solutions and their global and trajectory attractors. The results can be used for the investigation of specific physical models including combustion models in porous media, conduction models of electrical impulses into the nerve endings, climate models.

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References

  1. Balibrea, F., Caraballo, T., Kloeden, P.E., Valero, J.: Recent developments in dynamical systems: three perspectives. Int. J. Bifurcat. Chaos. 20, 2591 (2010). doi:10.1142/S0218127410027246

    Google Scholar 

  2. Ball, J.M.: Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. Nonlinear Sci. 7, 475–502 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ball, J.M.: Global attractors for damped semilinear wave equations. D.C.D.S. 10, 31–52 (2004)

    Google Scholar 

  4. Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. Nauka, Moscow (1989)

    MATH  Google Scholar 

  5. Chepyzhov, V.V., Vishik, M.I.: Evolution equations and their trajectory attractors. J. Math. Pur. Appl. 76(10), 913–964 (1997). doi:10.1016/S0021-7824(97)89978-3

    MathSciNet  MATH  Google Scholar 

  6. Chepyzhov, V.V., Vishik, M.I.: Trajectory and global attractors of three-dimensional Navier-Stokes systems. Math. Notes 71, 177–193 (2002). doi:10.1023/A:1014190629738

    Article  MathSciNet  MATH  Google Scholar 

  7. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    Google Scholar 

  8. Diestel, J.: Geometry of Banach Spaces, Selected Topics. Springer, New York (1980)

    MATH  Google Scholar 

  9. Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin (1974)

    MATH  Google Scholar 

  10. Gorban, N.V., Stanzhitsky, A.N.: On the dynamics of solutions for autonomous reaction-diffusion equation in \({\mathbb{R}}^N\) with multivalued nonlinearity. Ukr. Math. Bull. 6(2), 235–251 (2009)

    Google Scholar 

  11. Kapustyan, O.V., Valero, J.: Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions. Int. J. Bifur. Chaos. 20, 1–12 (2010). doi:10.1142/S0218127410027313

    Article  MathSciNet  Google Scholar 

  12. Kasyanov, P.O.: Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity. Cybern. Syst. Anal. 47(5), 800–811 (2011). doi:10.1007/s10559-011-9359-6

    Article  MathSciNet  Google Scholar 

  13. Kasyanov, P.O., Toscano, L., Zadoianchuk, N.V.: Regularity of weak solutions and their attractors for a parabolic feedback control problem. Set-Valued Var. Anal. 21(2), 271–282 (2013)

    Article  MathSciNet  Google Scholar 

  14. Ladyzhenskaya, O.A.: Dynamical system, generated by Navier-Stokes equations. Zap. Nauch. Sem. LOMI. 27, 91–115 (1972)

    MATH  Google Scholar 

  15. Ladyzhenskaya, O.A.: Some comments to my papers on the theory of attractors for abstract semigroups. Zap. Nauchn. Sem. LOMI. 188, 102–112 (1990)

    MathSciNet  Google Scholar 

  16. Ladyzhenskaya, O.A.: Attractors for Semigroups and Evolution Equations. University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  17. Lions, J.L.: Quelques M\(\acute{{\rm{e}}}\)thodes de R\(\acute{{\rm{e}}}\)solution des Probl\(\acute{{\rm{e}}}\)mes aux Limites non Lin\(\acute{{\rm{e}}}\)aires. Dunod, Paris (1969)

    Google Scholar 

  18. Melnik, V.S., Valero, J.: On attractors of multivalued semiflows and differential inclusions. Set Valued. Anal. 6, 83–111 (1998). doi:10.1023/A:1008608431399

    Article  MathSciNet  Google Scholar 

  19. Migórski, S., Ochal, A.: Optimal control of parabolic hemivariational inequalities. J. Global Optim. 17, 285–300 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Morillas, F., Valero, J.: Attractors for reaction-diffusion equation in \(\mathbb{R}^n\) with continuous nonlinearity. Asymptotic Anal. 44, 111–130 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Otani, M., Fujita, H.: On existence of strong solutions for \(\frac{{du}}{{dt}}(t)+\partial \varphi ^1(u(t))-\partial \varphi ^2(u(t))\ni f(t)\). J. Fac. Sci. Univ. Tokyo. Sect. 1. A 24(3), 575–605 (1977)

    Google Scholar 

  22. Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions. Birkhauser, Basel (1985)

    Book  MATH  Google Scholar 

  23. Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Springer, New York (2002)

    Book  MATH  Google Scholar 

  24. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)

    Book  MATH  Google Scholar 

  25. Valero, J.: Attractors of parabolic equations without uniqueness. J. Dyn. Diff. Equat. 13(4), 711–744 (2001). doi:10.1023/A:1016642525800

    Article  MathSciNet  MATH  Google Scholar 

  26. Valero, J., Kapustyan, A.V.: On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems. J. Math. Anal. Appl. 323, 614–633 (2006). doi:10.1016/j.jmaa.2005.10.042</doiurl>

    Article  MathSciNet  MATH  Google Scholar 

  27. Vishik, M.I., Zelik, S.V., Chepyzhov, V.V.: Strong trajectory attractor for dissipative reaction-diffusion system. Doclady Math. 82(3), 869–873 (2010). doi:10.1134/S1064562410060086

    Article  MathSciNet  MATH  Google Scholar 

  28. Zgurovsky, M.Z., Kasyanov, P.O., Valero, J.: Noncoercive evolution inclusions for Sk type operators. Int. J. Bifurcat. Chaos 20, 2823–2834 (2010). doi:10.1142/S0218127410027386

    Article  MathSciNet  MATH  Google Scholar 

  29. Zgurovsky, M.Z., Mel’nik, V.S., Kasyanov, P.O.: Evolution Inclusions and Variation Inequalities for Earth Data Processing I. Springer, Berlin (2010a). doi:10.1007/978-3-642-13837-9

  30. Zgurovsky, M.Z., Mel’nik V.S., Kasyanov P.O.: Evolution Inclusions and Variation Inequalities for Earth Data Processing II. Springer, Berlin (2010b). doi:10.1007/978-3-642-13878-2

  31. 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). doi:10.1007/978-3-642-28512-7

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Acknowledgments

The work was partially supported by the Ukrainian State Fund for Fundamental Researches under grant GP/F44/076.

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Authors and Affiliations

  1. Institute for Applied System Analysis, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Peremogy ave., 37, build, 35, Kyiv , 03056, Ukraine

    Nataliia V. Gorban & Pavlo O. Kasyanov

Authors
  1. Nataliia V. Gorban
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  2. Pavlo O. Kasyanov
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Correspondence to Nataliia V. Gorban .

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Editors and Affiliations

  1. National Academy of Science, National Technical University of Ukraine, Kiev, Ukraine

    Mikhail Z. Zgurovsky

  2. Lomonosov Moscow State University, Moscow, Russia

    Victor A. Sadovnichiy

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Gorban, N.V., Kasyanov, P.O. (2014). On Regularity of All Weak Solutions and Their Attractors for Reaction-Diffusion Inclusion in Unbounded Domain. In: Zgurovsky, M., Sadovnichiy, V. (eds) Continuous and Distributed Systems. Solid Mechanics and Its Applications, vol 211. Springer, Cham. https://doi.org/10.1007/978-3-319-03146-0_15

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  • DOI: https://doi.org/10.1007/978-3-319-03146-0_15

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  • Online ISBN: 978-3-319-03146-0

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