Regularity of Weak Solutions and Their Attractors for a Parabolic Feedback Control Problem

Pavlo O. Kasyanov1, Luisa Toscano2, Nina V. Zadoianchuk1
1Institute for Applied System Analysis, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Peremogy ave., 37, build, 35, 03056, Kyiv, Ukraine
2Dep. Math. and Appl., University of Naples “Federico II”, R. Caccioppoli, via Claudio 21, 80125, Naples, Italy

Tóm tắt

Từ khóa


Tài liệu tham khảo

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

Balibrea, F., Caraballo, T., Kloeden, P.E., Valero, J.: Recent developments in dynamical systems: three perspectives. Int. J. Bifur. Chaos 20(9), 2591–2630 (2010). doi: 10.1142/S0218127410027246

Ball, J.M.: Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations. Nonlinear Sci. 7, 475–502 (1997). Erratum, ibid 8:233,1998. Corrected version appears in ‘Mechanics: from Theory to Computation’. pp. 447–474. Springer Verlag (2000)

Ball, J.M.: Global attractors for damped semilinear wave equations. Discrete Contin. Dyn. Syst. 10, 31–52 (2004)

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

Chepyzhov, V.V., Vishik, M.I.: Trajectory and global attractors of three-dimensional Navier–Stokes systems. Math. Notes 71(1–2), 177–193 (2002). doi: 10.1023/A:1014190629738

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

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

Kalita, P.: Regularity and Rothe method error estimates for parabolic hemivariational inequality. J. Math. Anal. Appl. 389, 618–631 (2012)

Kalita, P.: Convergence of Rothe scheme for hemivariational inequalities of parabolic type. Int. J. Numer. Anal. Model 10, 445–465 (2013)

Kapustyan, O.V., Valero, J.: Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions. Int. J. Bifur. Chaos 20(9) 2723–2730 (2010). doi: 10.1142/S0218127410027313

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. 373(2), 535–547 (2011). doi: 10.1016/j.jmaa.2010.07.040

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

Kasyanov, P.O.: Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity. Math. Notes 92(1–2), 205–218 (2012). doi: 10.1134/S0001434612070231

Kas’yanov, P.O., Mel’nyk, V.S.: On properties of subdifferential mappings in Frechet spaces. Ukr. Math. J. 57(10), 1621–1634 (2005)

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, Article ID 450984, 21 p. (2012). doi: 10.1155/2012/450984

Ladyzhenskaya, O.A.: Dynamical system, generated by Navier–Stokes equations. Zap. Naucn. Semin. LOMI 27, 91–115 (1972)

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

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

Lions, J.L.: Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires. Dunod, Paris (1969)

Melnik, V.S., Valero, J.: On attractors of multivalued semiflows and differential inclusions. Set-Valued Anal. 6(1), 83–111 (1998). doi: 10.1023/A:1008608431399

Migórski, S.: On the existence of solutions for parabolic hemivariational inequalities. J. Comput. Appl. Math. 129, 77–87 (2001)

Migórski, S., Ochal, A.: Optimal Control of Parabolic Hemivariational Inequalities. J. Glob. Optim. 17, 285–300 (2000)

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. 1A, Math. 24(3), 575–605 (1977)

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

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

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

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

Valero, J., Kapustyan, A.V.: On the connectedness and asymptotic behaviour of solutions of reaction–diffusion systems. J. Math. Anal. Appl. 323(1), 614–633 (2006). doi: 10.1016/j.jmaa.2005.10.042

Vishik, M.I., Zelik, S.V., Chepyzhov, V.V.: Strong Trajectory Attractor for Dissipative Reaction-Diffusion System. Dokl. Math. 82(3), 869–873 (2010). doi: 10.1134/S1064562410060086

Zgurovsky, M.Z., Kasyanov, P.O., Valero, J.: Noncoercive evolution inclusions for Sk type operators. Int. J. Bifur. Chaos 20(9), 2823–2830 (2010). doi: 10.1142/S0218127410027386

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(10), 1569–1574 (2012). doi: 10.1016/j.aml.2012.01.016

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

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

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