Advertisement

Long-Time Behavior of State Functions for Badyko Models

  • Nataliia V. Gorban
  • Mark O. Gluzman
  • Pavlo O. Kasyanov
  • Alla M. Tkachuk
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 69)

Abstract

In this note we examine the long-time behavior of state functions for a climate energy balance model (Budyko Model) in the strongest topologies of the phase and the extended phase spaces. Strongest convergence results for all weak solutions are obtained. New structure and regularity properties for global and trajectory attractors are justified.

Keywords

Weak Solution Real Hilbert Space Regularity Property Rest Point Extended Phase Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

This work was partially supported by the Ukrainian State Fund for Fundamental Researches under grant GP/F66/14921 and by the National Academy of Sciences of Ukraine under grant 2284.

References

  1. 1.
    Arrieta, J.M., Rodrígues-Bernal, A., Valero, J.: Dynamics of a reaction-diffusion equation with discontinuous nonlinearity. Int. J. Bifurc. Chaos 16, 2695–2984 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aubin, T.: Nonlinear Analysis on Manifolds. Monge-Amp\({\acute{\rm {e}}}\)re Equations. Springer, Berlin (1980)Google Scholar
  3. 3.
    Balibrea, F., Caraballo, T., Kloeden, P.E., Valero, J.: Recent developments in dynamical systems: three perspectives. Int. J. Bifurc. Chaos (2010). doi: 10.1142/S0218127410027246
  4. 4.
    Budyko, M.I.: The effects of solar radiation variations on the climate of the Earth. Tellus 21, 611–619 (1969)CrossRefGoogle Scholar
  5. 5.
    Ball, J.M.: Global attractors for damped semilinear wave equations. DCDS 10, 31–52 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Editura Academiei, Bucuresti (1976)CrossRefzbMATHGoogle Scholar
  7. 7.
    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 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chepyzhov, V.V., Vishik, M.I.: Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discret. Contin. Dyn. Syst. Ser. A 27, 1493–1509 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chepyzhov, V.V., Conti, M., Pata, V.: A minimal approach to the theory of global attractors. Discret. Contin. Dyn. Syst. 32, 2079–2088 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D\({\acute{{\rm i}}}\)az, H., D\({\acute{{\rm i}}}\)az, J.: On a stochastic parabolic PDE arising in climatology. Rev. R. Acad. Cien. Serie A Mat. 96, 123–128 (2002)Google Scholar
  11. 11.
    D\({\acute{{\rm i}}}\)az, J., Tello, L.: Infinitely many stationary solutions for a simple climate model via a shooting method. Math. Methods Appl. Sci. 25, 327–334 (2002)Google Scholar
  12. 12.
    D\({\acute{{\rm i}}}\)az, J., Hern\({\acute{{\rm a}}}\)ndez, J., Tello, L.: On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology. J. Math. Anal. Appl. 216, 593–613 (1997)Google Scholar
  13. 13.
    D\({\acute{{\rm i}}}\)az, J., Hern\({\acute{{\rm a}}}\)ndez, J., Tello, L.: Some results about multiplicity and bifurcation of stationary solutions of a reaction diffusion climatological model. Rev. R. Acad. Cien. Serie A. Mat. 96(3), 357–366 (2002)Google Scholar
  14. 14.
    Feireisl, E., Norbury, J.: Some existence and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities. Proc. R. Soc. Edinb. A. 119(1–2), 1–17 (1991)Google Scholar
  15. 15.
    Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare operatorgleichungen und operatordifferentialgleichungen. Akademie-Verlag, Berlin (1974)zbMATHGoogle Scholar
  16. 16.
    Gluzman, M.O., Gorban, N.V., Kasyanov, P.O.: Lyapunov functions for differential inclusions and applications in physics, biology, and climatology. Continuous and distributed systems II. Theory and applications. Series studies in systems. Decis. Control 30, 233–243 (2015). doi: 10.1007/978-3-319-19075-4_14
  17. 17.
    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. (2015). https://dx.doi.ord/10.1016/j.aml.2014.08.006
  18. 18.
    Gluzman, M.O., Gorban, N.V., Kasyanov, P.O.: Lyapunov functions for weak solutions of reaction-diffusion equations with discontinuous interaction functions and its applications. Nonautonomous Dyn. Syst. (2015). doi: 10.1515/msds-2015-0001
  19. 19.
    Goldstein, G.R., Miranville, A.: A Cahn-Hilliard-Gurtin model with dynamic boundary conditions. Discret. Contin. Dyn. Syst. Ser. S 6, 387–400 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gorban, N.V., Kasyanov, P.O.: On regularity of all weak solutions and their attractors for reaction-diffusion inclusion in unbounded domai. Solid Mech. Appl. 211, 205–220 (2014)MathSciNetzbMATHGoogle Scholar
  21. 21.
    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). doi: 10.1016/j.na.2013.12.004 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gorban, N.V., Kapustyan, O.V., Kasyanov, P.O., Paliichuk, L.S.: On global attractors for autonomous damped wave equation with discontinuous nonlinearity. Solid Mech. Appl. 211, 221–237 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Efendiev, M., Miranville, A., Zelik, S.: Exponential attractors for a nonlinear reaction-diffusion system in \(R^3\). Comptes Rendus de l’Academie des Sciences-Series I - Mathematics 330, 713–718 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kalita, P., Lukaszewicz, G.: Global attractors for multivalued semiflows with weak continuity properties. Nonlinear Anal. Theory Methods Appl. 101, 124–143 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kalita, P., Lukaszewicz, G.: Attractors for Navier-Stokes flows with multivalued and nonmonotone subdifferential boundary conditions. Nonlinear Anal. Real World Appl. 19, 75–88 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kapustyan, O.V., Kasyanov, P.O., Valero, J.: Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Commun. Pure Appl. Anal. 13, 1891–1906 (2014). doi: 10.3934/cpaa.2014.13.1891 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kapustyan, O.V., Kasyanov, P.O., Valero, J.: Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Commun. Pure Appl. Anal. 34, 4155–4182 (2014). doi: 10.3934/dcds.2014.34.4155 MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kapustyan, O.V., Kasyanov, P.O., Valero, J., Zgurovsky, M.Z.: Structure of uniform global attractor for general non-autonomous reaction-diffusion system. Solid Mech. Appl. 211, 163–180 (2014)MathSciNetzbMATHGoogle Scholar
  29. 29.
    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, 271–282 (2013). doi: 10.1007/s11228-013-0233-8 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kasyanov, P.O.: Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity. Cybern. Syst. Anal. 47, 800–811 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kasyanov, P.O.: Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity. Math. Notes 92, 205–218 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kasyanov, P.O., Toscano, L., Zadoianchuk, N.V.: Long-time behaviour of solutions for autonomous evolution hemivariational inequality with multidimensional “reaction-displacement” law. Abstr. Appl. Anal. 2012, 21 (2012). doi: 10.1155/2012/450984
  33. 33.
    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 MathSciNetCrossRefGoogle Scholar
  34. 34.
    Mig\({\acute{{\rm o}}}\)rski, S.: On the existence of solutions for parabolic hemivariational inequalities. J. Comput. Appl. Math. 129, 77–87 (2001)Google Scholar
  35. 35.
    Mig\({\acute{{\rm o}}}\)rski, S., Ochal, A.: Optimal control of parabolic hemivariational inequalities. J. Glob. Optim. 17, 285–300 (2000)Google Scholar
  36. 36.
    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. The University of Tokyo. Sect. 1 A, Mathematics. 24(3), 575–605 (1977)Google Scholar
  37. 37.
    Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhauser, Basel (1985)CrossRefzbMATHGoogle Scholar
  38. 38.
    Sellers, W.D.: A global climatic model based on the energy balance of the Earth-atmosphere system. J. Appl. Meteorol. 8, 392–400 (1969)CrossRefGoogle Scholar
  39. 39.
    Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  40. 40.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)CrossRefzbMATHGoogle Scholar
  41. 41.
    Terman, D.: A free boundary problem arising from a bistable reaction diffusion equation. SIAM J. Math. Anal. 14, 1107–1129 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Terman, D.: A free boundary arising from a model for nerve conduction. J. Differ. Equ. 58(3), 345–363 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Valero, J.: Attractors of parabolic equations without uniqueness. J. Dyn. Differ. Equ. 13, 711–744 (2001). doi: 10.1023/A:1016642525800 MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Valero, J., Kapustyan, A.V.: On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems. J. Math. Anal. Appl. (2006). doi: 10.1016/j.jmaa.2005.10.042
  45. 45.
    Vishik, M.I., Zelik, S.V., Chepyzhov, V.V.: Strong trajectory attractor for dissipative reaction-diffusion system. Doclady Math. (2010). doi: 10.1134/S1064562410060086
  46. 46.
    Zadoianchuk, N.V., Kasyanov, P.O.: Dynamics of solutions of a class of second-order autonomous evolution inclusions. Cybern. Syst. Anal. 48, 414–428 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    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 CrossRefzbMATHGoogle Scholar
  48. 48.
    Zgurovsky, M.Z., Kasyanov, P.O.: Multivalued dynamics of solutions for autonomous operator differential equations in strongest topologies. Solid Mech. Appl. 211, 149–162 (2014)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Zgurovsky, M.Z., Kasyanov, P.O., Zadoianchuk, N.V.: Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem. Appl. Math. Lett. 25, 1569–1574 (2012). doi: 10.1016/j.aml.2012.01.016 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Nataliia V. Gorban
    • 1
  • Mark O. Gluzman
    • 2
  • Pavlo O. Kasyanov
    • 1
  • Alla M. Tkachuk
    • 3
  1. 1.Institute for Applied System AnalysisNational Technical University of Ukraine “Kyiv Polytechnic Institute”KyivUkraine
  2. 2.Department of Applied Physics and Applied MathematicsColumbia UniversityNew YorkUSA
  3. 3.Faculty of Automation and Computer SystemsNational University of Food TechnologiesKyivUkraine

Personalised recommendations