# American Institute of Mathematical Sciences

April  2020, 19(4): 2197-2217. doi: 10.3934/cpaa.2020096

## Chain recurrence and structure of $\omega$-limit sets of multivalued semiflows

 1 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine 2 Institute for Applied System Analysis, National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine 3 Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, 03202-Elche, Alicante, Spain

Dedicated to Prof. Tomás Caraballo on the occasion of his 60-th birthday

Received  July 2019 Revised  September 2019 Published  January 2020

Fund Project: The first two authors were partially supported by the State Fund for Fundamental Research of Ukraine. The third author was partially supported by Spanish Ministry of Economy and Competitiveness and FEDER, project MTM2016-74921-P, and by Spanish Ministry of Science, Innovation and Universities, project PGC2018-096540-B-I00.

We study properties of $\omega$-limit sets of multivalued semiflows like chain recurrence or the existence of cyclic chains.

First, we prove that under certain conditions the $\omega$-limit set of a trajectory is chain recurrent, applying this result to an evolution differential inclusion with upper semicontinous right-hand side.

Second, we give conditions ensuring that the $\omega$-limit set of a trajectory contains a cyclic chain. Using this result we are able to check that the $\omega$-limit set of every trajectory of a reaction-diffusion equation without uniqueness of solutions is an equilibrium.

Citation: Olexiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Chain recurrence and structure of $\omega$-limit sets of multivalued semiflows. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2197-2217. doi: 10.3934/cpaa.2020096
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##### References:
 [1] Ivan Gentil, Bogusław Zegarlinski. Asymptotic behaviour of reversible chemical reaction-diffusion equations. Kinetic & Related Models, 2010, 3 (3) : 427-444. doi: 10.3934/krm.2010.3.427 [2] Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155 [3] Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083 [4] Jacson Simsen, Mariza Stefanello Simsen, José Valero. Convergence of nonautonomous multivalued problems with large diffusion to ordinary differential inclusions. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2347-2368. doi: 10.3934/cpaa.2020102 [5] Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147 [6] Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581 [7] B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077 [8] Ricardo Enguiça, Andrea Gavioli, Luís Sanchez. A class of singular first order differential equations with applications in reaction-diffusion. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 173-191. doi: 10.3934/dcds.2013.33.173 [9] Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55 [10] C.B. Muratov. A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 867-892. doi: 10.3934/dcdsb.2004.4.867 [11] Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143 [12] Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343 [13] Peter E. Kloeden, Thomas Lorenz. Pullback attractors of reaction-diffusion inclusions with space-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1909-1964. doi: 10.3934/dcdsb.2017114 [14] Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493 [15] Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49 [16] Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2179-2199. doi: 10.3934/cpaa.2012.11.2179 [17] María Anguiano, P.E. Kloeden. Asymptotic behaviour of the nonautonomous SIR equations with diffusion. Communications on Pure & Applied Analysis, 2014, 13 (1) : 157-173. doi: 10.3934/cpaa.2014.13.157 [18] Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191 [19] Tianran Zhang. Traveling waves for a reaction-diffusion model with a cyclic structure. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1859-1870. doi: 10.3934/dcdsb.2020006 [20] Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229

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