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

[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.

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[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.

[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.

[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.

[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.

[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.

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