# American Institute of Mathematical Sciences

August  2019, 24(8): 3995-4020. doi: 10.3934/dcdsb.2019047

## Trajectory and global attractors for generalized processes

 1 Departamento de Matemática, Centro de Ciências Exatas e de Tecnologia, Universidade Federal de São Carlos, Caixa Postal 676, 13.565-905 São Carlos SP, Brazil 2 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas. Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain

Dedicated to Peter Kloeden on occasion if his 70th birthday

Received  September 2017 Revised  July 2018 Published  February 2019

Fund Project: This research was partially supported by Programa Ciência sem Fronteiras/CNPq process 200493/2015-9 and CNPq process 140943/2013-7, Ministério da Ciência e Tecnologia, Brazil, by FEDER and Ministerio de Economía y Competitividad under grant MTM2015-63723-P and Junta de Andalucía under Proyecto de Excelencia P12-FQM-1492

In this work the theory of generalized processes is used to describe the dynamics of a nonautonomous multivalued problem and, through this approach, some conditions for the existence of trajectory attractors are proved. By projecting the trajectory attractor on the phase space, the uniform attractor for the multivalued process associated to the problem is obtained and some conditions to guarantee the invariance of the uniform attractor are given. Furthermore, the existence of the uniform attractor for a class of $p$-Laplacian non-autonomous problems with dynamical boundary conditions is established.

Citation: Rodrigo Samprogna, Cláudia B. Gentile Moussa, Tomás Caraballo, Karina Schiabel. Trajectory and global attractors for generalized processes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3995-4020. doi: 10.3934/dcdsb.2019047
##### References:
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Ƚukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. Google Scholar [7] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. Google Scholar [8] A. N. Carvalho, J. A. Langa and J. C. Robinson, Non-autonomous dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 703-747. doi: 10.3934/dcdsb.2015.20.703. Google Scholar [9] V. Chepyzhov and M. I. Vishik, Trajectory attractors for the 2D Navier-Stokes system and some generalizations, Topol. Methods Nonlinear Anal., 8 (1996), 217-243. doi: 10.12775/TMNA.1996.030. Google Scholar [10] V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl. (9), 76 (1997), 913-964. doi: 10.1016/S0021-7824(97)89978-3. 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Google Scholar [16] C. G. Gal and M. Warma, Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions, Differential Integral Equations, 23 (2010), 327-358. Google Scholar [17] C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, J. Differential Equations, 253 (2012), 126-166. doi: 10.1016/j.jde.2012.02.010. Google Scholar [18] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. Google Scholar [19] T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43-60. doi: 10.1017/S0308210500023945. Google Scholar [20] J. Kacur, Nonlinear parabolic equations with the mixed nonlinear and nonstationary boundary conditions, Math. Slovaca, 30 (1980), 213-237. Google Scholar [21] O. V. Kapustyan and J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2723-2734. doi: 10.1142/S0218127410027313. Google Scholar [22] O. V. Kapustyan and J. Valero, On the Kneser property for the complex Ginzburg-Landau equation and the Lotka-Volterra system with diffusion, J. Math. Anal. Appl., 357 (2009), 254-272. doi: 10.1016/j.jmaa.2009.04.010. Google Scholar [23] O. V. Kapustyan, P. O. Kasyanov, J. Valero and M. Z. Zgurovsky, Structure of uniform global attractor for general non-autonomous reaction-diffusion system, Continuous and Distributed Systems, 163–180, Solid Mech. Appl., 211, Springer, Cham, 2014. doi: 10.1007/978-3-319-03146-0_12. Google Scholar [24] R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with fluid, Tohoku Math. J., 35 (1932), 260-275. Google Scholar [25] F. Li and B. You, Pullback attractor for non-autonomous p-Laplacian equations with dynamic flux boundary conditions, Electron. J. Differential Equations, 2014 (2014), No. 74, 11 pp. Google Scholar [26] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972. Google Scholar [27] H. W. MARCH and W. WEAVER, The diffusion problem for a solid in contact with a stirred fluid, Physical Review, 31 (1928), 1072-1082. Google Scholar [28] V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399. Google Scholar [29] V. S. Melnik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Anal., 8 (2000), 375-403. doi: 10.1023/A:1026514727329. Google Scholar [30] L. Popescu and A. Rodríguez-Bernal, On a singularly perturbed wave equation with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 389-413. doi: 10.1017/S0308210500003279. Google Scholar [31] J. C. Robinson, Infinite-dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0. Google Scholar [32] R. A. Samprogna, K. Schiabel and C. B. Gentile Moussa, Pullback attractors for multivalued process and application to nonautonomous problem with dynamic boundary conditions, Set-Valued and Variational Analysis: theory and applications, accepted for publication, 2017.Google Scholar [33] R. A. Samprogna, T. Caraballo, R. A. Samprogna and T. Caraballo, Pullback attractor for a dynamic boundary non-autonomous problem with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 509-523. doi: 10.3934/dcdsb.2017195. Google Scholar [34] J. Simsen and E. Capelato, Some properties for exact generalized processes. Continuous and Distributed Systems. II, 209–219, Stud. Syst. Decis. Control, 30, Springer, Cham, 2015. doi: 10.1007/978-3-319-19075-4_12. Google Scholar [35] J. Simsen and C. B. Gentile, On attractors for multivalued semigroups defined by generalized semiflows, Set-Valued Anal., 16 (2008), 105-124. doi: 10.1007/s11228-006-0037-1. Google Scholar [36] J. Valero and A. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems, J. Math. Anal. Appl., 323 (2006), 614-633. doi: 10.1016/j.jmaa.2005.10.042. Google Scholar [37] M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Lezioni Lincee. [Lincei Lectures] Cambridge University Press, Cambridge, 1992. Google Scholar [38] L. Yang, M. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2635-2651. doi: 10.3934/dcdsb.2012.17.2635. Google Scholar [39] L. Yang, M. Yang and J. Wu, On uniform attractors for non-autonomous p-Laplacian equation with a dynamic boundary condition, Topol. Methods Nonlinear Anal., 42 (2013), 169-180. Google Scholar

show all references

##### References:
 [1] J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502. doi: 10.1007/s003329900037. Google Scholar [2] J. M. Ball, On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations, J. Differential Equations, 27 (1978), 224-265. doi: 10.1016/0022-0396(78)90032-3. Google Scholar [3] R. F. Bass, Real Analysis for Graduate Students: Measure and Integration Theory, Copyright 2011 Richard F. Bass, 2011.Google Scholar [4] M. C. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, J. Differential Equations, 257 (2014), 490-522. doi: 10.1016/j.jde.2014.04.008. Google Scholar [5] T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201. doi: 10.1023/A:1022902802385. Google Scholar [6] T. Caraballo, G. Ƚukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. Google Scholar [7] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. Google Scholar [8] A. N. Carvalho, J. A. Langa and J. C. Robinson, Non-autonomous dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 703-747. doi: 10.3934/dcdsb.2015.20.703. Google Scholar [9] V. Chepyzhov and M. I. Vishik, Trajectory attractors for the 2D Navier-Stokes system and some generalizations, Topol. Methods Nonlinear Anal., 8 (1996), 217-243. doi: 10.12775/TMNA.1996.030. Google Scholar [10] V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl. (9), 76 (1997), 913-964. doi: 10.1016/S0021-7824(97)89978-3. Google Scholar [11] V. Chepyzhov and M. I, Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002. Google Scholar [12] V. Chepyzhov and M. I. Vishik, Trajectory and global attractors of the three-dimensional Navier-Stokes system, Math. Notes, 71 (2002), 177-193. doi: 10.1023/A:1014190629738. Google Scholar [13] A. Cheskidov and L. Kavlie, Pullback attractors for generalized evolutionary systems, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 749-779. doi: 10.3934/dcdsb.2015.20.749. Google Scholar [14] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976. Google Scholar [15] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19. doi: 10.1007/s00028-002-8077-y. Google Scholar [16] C. G. Gal and M. Warma, Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions, Differential Integral Equations, 23 (2010), 327-358. Google Scholar [17] C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, J. Differential Equations, 253 (2012), 126-166. doi: 10.1016/j.jde.2012.02.010. Google Scholar [18] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. Google Scholar [19] T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 113 (1989), 43-60. doi: 10.1017/S0308210500023945. Google Scholar [20] J. Kacur, Nonlinear parabolic equations with the mixed nonlinear and nonstationary boundary conditions, Math. Slovaca, 30 (1980), 213-237. Google Scholar [21] O. V. Kapustyan and J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2723-2734. doi: 10.1142/S0218127410027313. Google Scholar [22] O. V. Kapustyan and J. Valero, On the Kneser property for the complex Ginzburg-Landau equation and the Lotka-Volterra system with diffusion, J. Math. Anal. Appl., 357 (2009), 254-272. doi: 10.1016/j.jmaa.2009.04.010. Google Scholar [23] O. V. Kapustyan, P. O. Kasyanov, J. Valero and M. Z. Zgurovsky, Structure of uniform global attractor for general non-autonomous reaction-diffusion system, Continuous and Distributed Systems, 163–180, Solid Mech. Appl., 211, Springer, Cham, 2014. doi: 10.1007/978-3-319-03146-0_12. Google Scholar [24] R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with fluid, Tohoku Math. J., 35 (1932), 260-275. Google Scholar [25] F. Li and B. You, Pullback attractor for non-autonomous p-Laplacian equations with dynamic flux boundary conditions, Electron. J. Differential Equations, 2014 (2014), No. 74, 11 pp. Google Scholar [26] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972. Google Scholar [27] H. W. MARCH and W. WEAVER, The diffusion problem for a solid in contact with a stirred fluid, Physical Review, 31 (1928), 1072-1082. Google Scholar [28] V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399. Google Scholar [29] V. S. Melnik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Anal., 8 (2000), 375-403. doi: 10.1023/A:1026514727329. Google Scholar [30] L. Popescu and A. Rodríguez-Bernal, On a singularly perturbed wave equation with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 389-413. doi: 10.1017/S0308210500003279. Google Scholar [31] J. C. Robinson, Infinite-dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0. Google Scholar [32] R. A. Samprogna, K. Schiabel and C. B. Gentile Moussa, Pullback attractors for multivalued process and application to nonautonomous problem with dynamic boundary conditions, Set-Valued and Variational Analysis: theory and applications, accepted for publication, 2017.Google Scholar [33] R. A. Samprogna, T. Caraballo, R. A. Samprogna and T. Caraballo, Pullback attractor for a dynamic boundary non-autonomous problem with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 509-523. doi: 10.3934/dcdsb.2017195. Google Scholar [34] J. Simsen and E. Capelato, Some properties for exact generalized processes. Continuous and Distributed Systems. II, 209–219, Stud. Syst. Decis. Control, 30, Springer, Cham, 2015. doi: 10.1007/978-3-319-19075-4_12. Google Scholar [35] J. Simsen and C. B. Gentile, On attractors for multivalued semigroups defined by generalized semiflows, Set-Valued Anal., 16 (2008), 105-124. doi: 10.1007/s11228-006-0037-1. Google Scholar [36] J. Valero and A. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems, J. Math. Anal. Appl., 323 (2006), 614-633. doi: 10.1016/j.jmaa.2005.10.042. Google Scholar [37] M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Lezioni Lincee. [Lincei Lectures] Cambridge University Press, Cambridge, 1992. Google Scholar [38] L. Yang, M. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2635-2651. doi: 10.3934/dcdsb.2012.17.2635. Google Scholar [39] L. Yang, M. Yang and J. Wu, On uniform attractors for non-autonomous p-Laplacian equation with a dynamic boundary condition, Topol. Methods Nonlinear Anal., 42 (2013), 169-180. Google Scholar
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