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
References:
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E. Aragão-CostaT. CaraballoA. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbations, Nonlinearity, 24 (2011), 2099-2117.  doi: 10.1088/0951-7715/24/7/010.  Google Scholar

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J. ArrietaA. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, Internat. J. Bifur. Chaos, 16 (2006), 2965-2984.  doi: 10.1142/S0218127406016586.  Google Scholar

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J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäusser, 1990.  Google Scholar

[4]

A. V. Babin and M. I. Vishik, Maximal attractors of semigroups corresponding to evolutionary differential equations, Mat. Sb., 126 (1985), 397–419.  Google Scholar

[5]

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

[6]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[7]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976.  Google Scholar

[8]

G. Butler and P. Waltman, Persistence in dynamical systems, J. Differential Equations, 63 (1986), 255-263.  doi: 10.1016/0022-0396(86)90049-5.  Google Scholar

[9]

R. Caballero, A. N. Carvalho, P. Marín-Rubio and J. Valero, Robustness of dynamically gradient multivalued dynamical systems, Discrete Contin. Dyn. Syst., Series B, 24 (2019), 1049–1077.  Google Scholar

[10]

T. CaraballoJ. A. Langa and J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal., 48 (2002), 805-829.  doi: 10.1016/S0362-546X(00)00216-9.  Google Scholar

[11]

T. CaraballoJ. A. Langa and J. Valero, Asymptotic behaviour of monotone multi-valued dynamical systems, Dynam. Syst., 20 (2005), 301-321.  doi: 10.1080/14689360500151847.  Google Scholar

[12]

T. CaraballoJ. A. Langa and J. Valero, Structure of the pullback attractor for a non-autonomous scalar differential inclusion, Discrete Contin. Dyn. Syst., Series S, 9 (2016), 979-994.  doi: 10.3934/dcdss.2016037.  Google Scholar

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T. Caraballo, J. A. Langa and J. Valero, Extremal bounded complete trajectories for nonautonomous reaction-diffusion equations with discontinuous forcing term, Revista Matemática Complutense, to appear. Google Scholar

[14]

T. CaraballoP. Marin-Rubio and J. Robinson, A comparison between two theories for multivalued semiflows and their asymptotic behavior, Set-valued Anal., 11 (2003), 297-322.  doi: 10.1023/A:1024422619616.  Google Scholar

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

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V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 2002.  Google Scholar

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C. Conley, The gradient structure of a flow. I, Ergodic Theory Dynam. Systems, 8 (1988), 11-26.  doi: 10.1017/S0143385700009305.  Google Scholar

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H. B. da Costa and J. Valero, Morse decompositions and Lyapunov functions for dynamically gradient multivalued semiflows, Nonlinear Dyn., 84 (2016), 19-34.  doi: 10.1007/s11071-015-2193-z.  Google Scholar

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H. B. da Costa and J. Valero, Morse decompositions with infinite components for multivalued semiflows, Set-Valued Var. Anal., 25 (2017), 25-41.  doi: 10.1007/s11228-016-0363-x.  Google Scholar

[20]

Z. Denkowski and S. Mortola, Asymptotic behavior of optimal solutions to control problems for systems described by differential inclusions corresponding to partial differential equations, Journal of Optimization Theory and Applications, 78 (1993), 365-391.  doi: 10.1007/BF00939675.  Google Scholar

[21]

N. V. Gorban, O. V. Kapustyan, P. O. Kasyanov and L. S. Palichuk, On global attractors for autonomous damped wave equation with discontinuous nonlinearity, in, Continuous and Distributed Systems (M.Z. Zgurovsky and V.A. Sadovnichiy eds.), vol. 211, pp. 221-237, Cham, Springer (2014). doi: 10.1007/978-3-319-03146-0_16.  Google Scholar

[22]

P. Kalita and G. Lukaszewich, Global attractors for multivalued semiflows with weak continuity properties, Nonlinear Anal., 101 (2014), 124-143.  doi: 10.1016/j.na.2014.01.026.  Google Scholar

[23]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025.  Google Scholar

[24]

M. Hurley, Chain reccurence, semiflows and gradients, J. Dynamics Differential Equations, 7 (1995), 437-456.  doi: 10.1007/BF02219371.  Google Scholar

[25]

A. Kapustyan, Global attractors for nonautonomous reaction-diffusion equation, Differ. Equ., 10 (2002), 1378-1382.  doi: 10.1023/A:1022378831393.  Google Scholar

[26]

O. V. KapustyanA. V. Pankov and J. Valero, On global attractors of multivalued semiflows generated by the 3D Bénard system, Set-Valued Var. Anal., 20 (2012), 445-465.  doi: 10.1007/s11228-011-0197-5.  Google Scholar

[27]

O. V. Kapustyan and J. Valero, Attractors of differential inclusions and their approximation, Ukrain. Mat. Zh., 52 (2000), 975–979 (translated in Ukrainian Math. J., 52 (2000), 1118–1123). doi: 10.1023/A:1005237902620.  Google Scholar

[28]

O. V. KapustyanP. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term, Discrete Contin. Dyn. Syst., Series A, 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.  Google Scholar

[29]

O. V. Kapustyan, P.O. Kasyanov and J. Valero, Structure of the global attractor for weak solutions of a reaction-diffusion equation, Appl. Math. Inf. Sci., 9 (2015), 2257–2264.  Google Scholar

[30]

P. O. Kasyanov, Multivalued dynamics of solutions of autonomous differential-operator inclusion with pseudomonotone nonlinearity, Cybernet. Systems Anal., 47 (2011), 800-811.  doi: 10.1007/s10559-011-9359-6.  Google Scholar

[31]

P. E. Kloeden and J. Valero, Attractors of weakly asymptotically compact set-valued dynamical systems, Set-Valued Anal., 13 (2005), 381-404.  doi: 10.1007/s11228-004-0047-9.  Google Scholar

[32]

O. V. Kapustyan, V. S. Melnik, J. Valero and V. V. Yasinsky, Global Attractors of Multivalued Dynamical Systems and Evolution Equations without Uniqueness, Naukova Dumka, Kyiv, 2008. Google Scholar

[33]

D. Li, Morse decompositions for general dynamical systems and differential inclusions with applications to control systems, SIAM J. Control Optim., 46 (2007), 35-60.  doi: 10.1137/060662101.  Google Scholar

[34]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83–111. Google Scholar

[35]

K. MischaikowH. Smith and H. R. Thieme, Asymptoticaly autonomous semiflows: chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.  doi: 10.2307/2154964.  Google Scholar

[36]

J. Simsen and C. 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

[37]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[38]

A. A. Tostonogov and Y. I. Umanskiy, On solutions of evolution inclusions. II, Sibir. Math. J., 33 (1992), 163-174.  doi: 10.1007/BF00971135.  Google Scholar

[39]

M. Z. Zgurovsky and P. O. Kasyanov, Qualitative and Quantitative Analysis of Nonlinear Systems: Theory and Applications, Series: Studies in Systems, Decision and Control, Vol. 111, Cham, Springer, 2018.  Google Scholar

show all references

References:
[1]

E. Aragão-CostaT. CaraballoA. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbations, Nonlinearity, 24 (2011), 2099-2117.  doi: 10.1088/0951-7715/24/7/010.  Google Scholar

[2]

J. ArrietaA. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, Internat. J. Bifur. Chaos, 16 (2006), 2965-2984.  doi: 10.1142/S0218127406016586.  Google Scholar

[3]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäusser, 1990.  Google Scholar

[4]

A. V. Babin and M. I. Vishik, Maximal attractors of semigroups corresponding to evolutionary differential equations, Mat. Sb., 126 (1985), 397–419.  Google Scholar

[5]

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

[6]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[7]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976.  Google Scholar

[8]

G. Butler and P. Waltman, Persistence in dynamical systems, J. Differential Equations, 63 (1986), 255-263.  doi: 10.1016/0022-0396(86)90049-5.  Google Scholar

[9]

R. Caballero, A. N. Carvalho, P. Marín-Rubio and J. Valero, Robustness of dynamically gradient multivalued dynamical systems, Discrete Contin. Dyn. Syst., Series B, 24 (2019), 1049–1077.  Google Scholar

[10]

T. CaraballoJ. A. Langa and J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal., 48 (2002), 805-829.  doi: 10.1016/S0362-546X(00)00216-9.  Google Scholar

[11]

T. CaraballoJ. A. Langa and J. Valero, Asymptotic behaviour of monotone multi-valued dynamical systems, Dynam. Syst., 20 (2005), 301-321.  doi: 10.1080/14689360500151847.  Google Scholar

[12]

T. CaraballoJ. A. Langa and J. Valero, Structure of the pullback attractor for a non-autonomous scalar differential inclusion, Discrete Contin. Dyn. Syst., Series S, 9 (2016), 979-994.  doi: 10.3934/dcdss.2016037.  Google Scholar

[13]

T. Caraballo, J. A. Langa and J. Valero, Extremal bounded complete trajectories for nonautonomous reaction-diffusion equations with discontinuous forcing term, Revista Matemática Complutense, to appear. Google Scholar

[14]

T. CaraballoP. Marin-Rubio and J. Robinson, A comparison between two theories for multivalued semiflows and their asymptotic behavior, Set-valued Anal., 11 (2003), 297-322.  doi: 10.1023/A:1024422619616.  Google Scholar

[15]

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

[16]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 2002.  Google Scholar

[17]

C. Conley, The gradient structure of a flow. I, Ergodic Theory Dynam. Systems, 8 (1988), 11-26.  doi: 10.1017/S0143385700009305.  Google Scholar

[18]

H. B. da Costa and J. Valero, Morse decompositions and Lyapunov functions for dynamically gradient multivalued semiflows, Nonlinear Dyn., 84 (2016), 19-34.  doi: 10.1007/s11071-015-2193-z.  Google Scholar

[19]

H. B. da Costa and J. Valero, Morse decompositions with infinite components for multivalued semiflows, Set-Valued Var. Anal., 25 (2017), 25-41.  doi: 10.1007/s11228-016-0363-x.  Google Scholar

[20]

Z. Denkowski and S. Mortola, Asymptotic behavior of optimal solutions to control problems for systems described by differential inclusions corresponding to partial differential equations, Journal of Optimization Theory and Applications, 78 (1993), 365-391.  doi: 10.1007/BF00939675.  Google Scholar

[21]

N. V. Gorban, O. V. Kapustyan, P. O. Kasyanov and L. S. Palichuk, On global attractors for autonomous damped wave equation with discontinuous nonlinearity, in, Continuous and Distributed Systems (M.Z. Zgurovsky and V.A. Sadovnichiy eds.), vol. 211, pp. 221-237, Cham, Springer (2014). doi: 10.1007/978-3-319-03146-0_16.  Google Scholar

[22]

P. Kalita and G. Lukaszewich, Global attractors for multivalued semiflows with weak continuity properties, Nonlinear Anal., 101 (2014), 124-143.  doi: 10.1016/j.na.2014.01.026.  Google Scholar

[23]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025.  Google Scholar

[24]

M. Hurley, Chain reccurence, semiflows and gradients, J. Dynamics Differential Equations, 7 (1995), 437-456.  doi: 10.1007/BF02219371.  Google Scholar

[25]

A. Kapustyan, Global attractors for nonautonomous reaction-diffusion equation, Differ. Equ., 10 (2002), 1378-1382.  doi: 10.1023/A:1022378831393.  Google Scholar

[26]

O. V. KapustyanA. V. Pankov and J. Valero, On global attractors of multivalued semiflows generated by the 3D Bénard system, Set-Valued Var. Anal., 20 (2012), 445-465.  doi: 10.1007/s11228-011-0197-5.  Google Scholar

[27]

O. V. Kapustyan and J. Valero, Attractors of differential inclusions and their approximation, Ukrain. Mat. Zh., 52 (2000), 975–979 (translated in Ukrainian Math. J., 52 (2000), 1118–1123). doi: 10.1023/A:1005237902620.  Google Scholar

[28]

O. V. KapustyanP. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term, Discrete Contin. Dyn. Syst., Series A, 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.  Google Scholar

[29]

O. V. Kapustyan, P.O. Kasyanov and J. Valero, Structure of the global attractor for weak solutions of a reaction-diffusion equation, Appl. Math. Inf. Sci., 9 (2015), 2257–2264.  Google Scholar

[30]

P. O. Kasyanov, Multivalued dynamics of solutions of autonomous differential-operator inclusion with pseudomonotone nonlinearity, Cybernet. Systems Anal., 47 (2011), 800-811.  doi: 10.1007/s10559-011-9359-6.  Google Scholar

[31]

P. E. Kloeden and J. Valero, Attractors of weakly asymptotically compact set-valued dynamical systems, Set-Valued Anal., 13 (2005), 381-404.  doi: 10.1007/s11228-004-0047-9.  Google Scholar

[32]

O. V. Kapustyan, V. S. Melnik, J. Valero and V. V. Yasinsky, Global Attractors of Multivalued Dynamical Systems and Evolution Equations without Uniqueness, Naukova Dumka, Kyiv, 2008. Google Scholar

[33]

D. Li, Morse decompositions for general dynamical systems and differential inclusions with applications to control systems, SIAM J. Control Optim., 46 (2007), 35-60.  doi: 10.1137/060662101.  Google Scholar

[34]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83–111. Google Scholar

[35]

K. MischaikowH. Smith and H. R. Thieme, Asymptoticaly autonomous semiflows: chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.  doi: 10.2307/2154964.  Google Scholar

[36]

J. Simsen and C. 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

[37]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[38]

A. A. Tostonogov and Y. I. Umanskiy, On solutions of evolution inclusions. II, Sibir. Math. J., 33 (1992), 163-174.  doi: 10.1007/BF00971135.  Google Scholar

[39]

M. Z. Zgurovsky and P. O. Kasyanov, Qualitative and Quantitative Analysis of Nonlinear Systems: Theory and Applications, Series: Studies in Systems, Decision and Control, Vol. 111, Cham, Springer, 2018.  Google Scholar

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