Abstract
In this chapter we consider the problem of uniform convergence results for all globally defined weak solutions of non-autonomous reaction-diffusion system with Carathéodory’s nonlinearity satisfying standard sign and polynomial growth assumptions. The main contributions of this chapter are: the uniform convergence results for all globally defined weak solutions of non-autonomous reaction-diffusion equations with Carathéodory’s nonlinearity and sufficient conditions for the convergence of weak solutions in strongest topologies.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. Nauka, Moscow (1989). [in Russian]
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
Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare operatorgleichungen und operatordifferentialgleichungen. Akademie-Verlag, Berlin (1978)
Chepyzhov V.V., Vishik M.I.: Trajectory attractors for evolution equations. C.R.Acad.Sci. Paris. Serie I, vol. 321, pp. 1309–1314 (1995)
Chepyzhov, V.V., Vishik, M.I.: Evolution equations and their trajectory attractors. J. Math. Pures Appl. 76, 913–964 (1997)
Chepyzhov, V.V., Vishik, M.I.: Trajectory and global attractors for 3D Navier-Stokes system. Mat. Zametki. (2002). doi:10.1023/A:1014190629738
Chepyzhov, V.V., Vishik, M.I.: Attractors for equations of mathematical physics. American Mathematical Society, Providence RI (2002)
Chepyzhov, V.V., Vishik, M.I.: Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discret. Contin. Dyn. Syst. 27(4), 1498–1509 (2010)
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
Hale, J.K.: Asymptotic Behavior of Dissipative Systems. AMS, Providence, RI (1988)
Kasyanov, P.O.: Multi-valued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity. Mathematical Notes 92, 205–218 (2012)
Kasyanov, P.O.: Multi-valued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity. Cybern. Syst. Anal. 47, 800–811 (2011)
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. (2013). doi:10.1007/s11228-013-0233-8
Kapustyan, O.V., Melnik, V.S., Valero, J.: Attractors of multi-valued dynamical processes, generated by phase-field equations. Int. J. Bifurc. Chaos 6, 34–45 (2003)
Kapustyan, O.V., Kasyanov, P.O., Valero, J.: Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes equations. J. Math. Anal. Appl. (2011). doi:10.1016/j.jmaa.2010.07.040
Kapustyan, A.V., Valero, J.: On the Kneser property for the complex Ginzburg-Landau equation and the Lotka-Volterra system with diffusion. J. Math. Anal. Appl. 357, 254–272 (2009)
Krasnosel’skii, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. International Series of Monographs on Pure and Applied Mathematics, vol. 45, pp. X + 395 S. Oxford (1964)
Ladyzhenskaya, O.A.: Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)
Melnik, V.S., Valero, J.: On attractors of multi-valued semi-flows and generalized differential equations. Set-Valued Anal. 6(1), 83–111 (1998)
Mel’nik, V.S., Valero J.: On global attractors of multi-valued semiprocesses and nonautonomous evolution inclusions. Set-Valued Anal. (2000). doi:10.1023/A:1026514727329
Sell, G.R.: Global attractors for the three-dimensional Navier-Stokes equations. J. Dyn. Diff. Eq. 8(12), 1–33 (1996)
Smoller, J.: Shock Waves and Reaction-Diffusion Equations (Grundlehren der mathematischen Wissenschaften). Springer, New York (1983)
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 Appl. Math. Sci. Springer, New York (1988)
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
Vishik, M.I., Zelik, S.V., Chepyzhov, V.V.: Strong trajectory attractor for a dissipative reaction-diffusion system. Doklady Math. 82(3), 869–873 (2010)
Warga J.: Optimal Control of Differential and Functional Equations. Academic Press, Cambridge (1972)
Zgurovsky, M.Z., Mel’nik, V.S., Kasyanov, P.O.: Evolution Inclusions and Variation Inequalities for Earth Data Processing II. Springer, Berlin (2011)
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)
Zgurovsky, M.Z., Kasyanov, P.O.: Zadoianchuk (Zadoyanchuk) N.V.: Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem. Appl. Math. Lett. 25(10), 1569–1574 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Zgurovsky, M.Z., Kasyanov, P.O. (2018). Strongest Convergence Results for Weak Solutions of Non-autonomous Reaction-Diffusion Equations with Carathéodory’s Nonlinearity. In: Qualitative and Quantitative Analysis of Nonlinear Systems. Studies in Systems, Decision and Control, vol 111. Springer, Cham. https://doi.org/10.1007/978-3-319-59840-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-59840-6_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-59839-0
Online ISBN: 978-3-319-59840-6
eBook Packages: EngineeringEngineering (R0)