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July  2017, 22(5): 1875-1886. doi: 10.3934/dcdsb.2017111

Global attractors of impulsive parabolic inclusions

Sergey Dashkovskiy 1, , Oleksiy Kapustyan 2,, and  Iryna Romaniuk 2,

1. 

Institute of Mathematics, University of Würzburg, Emil-Fischer-Straße 40, Würzburg, Germany
2. 

Taras Shevchenko National University of Kyiv, Department of Mathematics and Mechanics, Volodymyrska Str. 60,01033, Kyiv, Ukraine

* Corresponding author: O. Kapustyan

This work is supported by the German Research Foundation (DFG) via grant DA 767/8-1 The second author is also supported by the State Fund For Fundamental Research, Grant of President of Ukraine, Project F62/94-2015.

Received  November 2015 Revised  March 2016 Published  March 2017

In this work we consider an impulsive multi-valued dynamical system generated by a parabolic inclusion with upper semicontinuous right-hand side $\varepsilon F(y)$ and with impulsive multi-valued perturbations. Moments of impulses are not fixed and defined by moments of intersection of solutions with some subset of the phase space. We prove that for sufficiently small value of the parameter $\varepsilon>0$ this system has a global attractor.

Keywords: Global attractor, multi-valued dynamical system, impulsive dynamical system, parabolic inclusion.
Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 35K55, 58C0.
Citation: Sergey Dashkovskiy, Oleksiy Kapustyan, Iryna Romaniuk. Global attractors of impulsive parabolic inclusions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1875-1886. doi: 10.3934/dcdsb.2017111
References:
[1]

M. U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Anal., 60 (2005), 163-178.  doi: 10.1016/S0362-546X(04)00347-5.  Google Scholar

[2]

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

[3]

E. M. Bonotto, Flows of characteristic 0+ in impulsive semidynamical systems, J. Math. Anal. Appl., 332 (2007), 81-96.  doi: 10.1016/j.jmaa.2006.09.076.  Google Scholar

[4]

E. M. Bonotto and D. P. Demuner, Attractors of impulsive dissipative semidynamical systems, Bull. Sci. Math., 137 (2013), 617-642.  doi: 10.1016/j.bulsci.2012.12.005.  Google Scholar

[5]

E. M. Bonotto, M. C. Bortolan, A. N. Carvalho and R. Czaja, Global attractors for impulsive dynamical systems -a precompact approach, J. Diff. Eqn., 259 (2015), 2602-2625.  doi: 10.1016/j.jde.2015.03.033.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors of Equations of Mathematical Physics AMS, 2002. Google Scholar

[7]

K. Ciesielski, On stability in impulsive dynamical systems, Bull. Pol. Acad. Sci. Math., 52 (2004), 81-91.  doi: 10.4064/ba52-1-9.  Google Scholar

[8]

S. Dachkovski, Anisotropic function spaces and related semi-linear hypoelliptic equations, Math. Nachr., 248 (2003), 40-61.  doi: 10.1002/mana.200310002.  Google Scholar

[9]

S. Dashkovskiy and A. Mironchenko, Input-to-state stability of nonlinear impulsive systems, SIAM J. Control. Optim., 51 (2013), 1962-1987.  doi: 10.1137/120881993.  Google Scholar

[10]

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

[11]

G. Iovane, O. V. Kapustyan and J. Valero, Asymptotic behaviour of reaction-diffusion equations with non-damped impulsive effects, Nonlinear Analysis, 68 (2008), 2516-2530.  doi: 10.1016/j.na.2007.02.002.  Google Scholar

[12]

A. V. Kapustyan and V. S. Mel'nik, On global attractors of multivalued semidynamical systems and their approximations, Doklady Academii Nauk., 366 (1999), 445-448.   Google Scholar

[13]

O. V. Kapustyan and D. V. Shkundin, Global attractor of one nonlinear parabolic equitation, Ukrainian Math. J., 55 (2003), 446-455.  doi: 10.1023/B:UKMA.0000010155.48722.f2.  Google Scholar

[14]

O. V. Kapustyan, V. S. Melnik and J. Valero, A weak attractor and properties of solutions for the three-dimensional Benard problem, Discrete and Continuous Dynamical Systems, 18 (2007), 449-481.  doi: 10.3934/dcds.2007.18.449.  Google Scholar

[15]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Pullback attractors for some class of extremal solutions of 3D Navier-Stokes system, J. Math. Anal. Appl., 373 (2011), 535-547.  doi: 10.1016/j.jmaa.2010.07.040.  Google Scholar

[16]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term, Discrete and continuous dynamical systems, 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.  Google Scholar

[17]

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

[18]

S. K. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.  doi: 10.1016/0022-247X(90)90199-P.  Google Scholar

[19]

S. K. Kaul, Stability and asymptotic stability in impulsive semidynamical systems, J. Appl. Math. Stoch. Anal., 7 (1994), 509-523.  doi: 10.1155/S1048953394000390.  Google Scholar

[20]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[21]

Y. M. Perestjuk, Discontinuous oscillations in an impulsive system, J. Math. Sci., 194 (2013), 404-413.  doi: 10.1007/s10958-013-1536-x.  Google Scholar

[22]

V. Rozko, Stability in terms of Lyapunov of discontinuous dynamic systems, Differ.Uravn., 11 (1975), 1005-1012.   Google Scholar

[23]

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equitations Singapore : World Scientific, 1995. Google Scholar

[24]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer, New York, 1988. Google Scholar

[25]

J. Valero, Finite and infinite-dimensional attractors of multivalued reaction-diffusion equations, Acta Mathematica Hungar., 88 (2000), 239-258.  doi: 10.1023/A:1006769315268.  Google Scholar

[26]

M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and N. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing Ⅲ. Long-Time Behaviour of Evolution Inclusions Solutions in Earth Data Analysis Springer, Heidelberg, 2012. Google Scholar

show all references

References:
[1]

M. U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Anal., 60 (2005), 163-178.  doi: 10.1016/S0362-546X(04)00347-5.  Google Scholar

[2]

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

[3]

E. M. Bonotto, Flows of characteristic 0+ in impulsive semidynamical systems, J. Math. Anal. Appl., 332 (2007), 81-96.  doi: 10.1016/j.jmaa.2006.09.076.  Google Scholar

[4]

E. M. Bonotto and D. P. Demuner, Attractors of impulsive dissipative semidynamical systems, Bull. Sci. Math., 137 (2013), 617-642.  doi: 10.1016/j.bulsci.2012.12.005.  Google Scholar

[5]

E. M. Bonotto, M. C. Bortolan, A. N. Carvalho and R. Czaja, Global attractors for impulsive dynamical systems -a precompact approach, J. Diff. Eqn., 259 (2015), 2602-2625.  doi: 10.1016/j.jde.2015.03.033.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors of Equations of Mathematical Physics AMS, 2002. Google Scholar

[7]

K. Ciesielski, On stability in impulsive dynamical systems, Bull. Pol. Acad. Sci. Math., 52 (2004), 81-91.  doi: 10.4064/ba52-1-9.  Google Scholar

[8]

S. Dachkovski, Anisotropic function spaces and related semi-linear hypoelliptic equations, Math. Nachr., 248 (2003), 40-61.  doi: 10.1002/mana.200310002.  Google Scholar

[9]

S. Dashkovskiy and A. Mironchenko, Input-to-state stability of nonlinear impulsive systems, SIAM J. Control. Optim., 51 (2013), 1962-1987.  doi: 10.1137/120881993.  Google Scholar

[10]

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

[11]

G. Iovane, O. V. Kapustyan and J. Valero, Asymptotic behaviour of reaction-diffusion equations with non-damped impulsive effects, Nonlinear Analysis, 68 (2008), 2516-2530.  doi: 10.1016/j.na.2007.02.002.  Google Scholar

[12]

A. V. Kapustyan and V. S. Mel'nik, On global attractors of multivalued semidynamical systems and their approximations, Doklady Academii Nauk., 366 (1999), 445-448.   Google Scholar

[13]

O. V. Kapustyan and D. V. Shkundin, Global attractor of one nonlinear parabolic equitation, Ukrainian Math. J., 55 (2003), 446-455.  doi: 10.1023/B:UKMA.0000010155.48722.f2.  Google Scholar

[14]

O. V. Kapustyan, V. S. Melnik and J. Valero, A weak attractor and properties of solutions for the three-dimensional Benard problem, Discrete and Continuous Dynamical Systems, 18 (2007), 449-481.  doi: 10.3934/dcds.2007.18.449.  Google Scholar

[15]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Pullback attractors for some class of extremal solutions of 3D Navier-Stokes system, J. Math. Anal. Appl., 373 (2011), 535-547.  doi: 10.1016/j.jmaa.2010.07.040.  Google Scholar

[16]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term, Discrete and continuous dynamical systems, 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.  Google Scholar

[17]

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

[18]

S. K. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.  doi: 10.1016/0022-247X(90)90199-P.  Google Scholar

[19]

S. K. Kaul, Stability and asymptotic stability in impulsive semidynamical systems, J. Appl. Math. Stoch. Anal., 7 (1994), 509-523.  doi: 10.1155/S1048953394000390.  Google Scholar

[20]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[21]

Y. M. Perestjuk, Discontinuous oscillations in an impulsive system, J. Math. Sci., 194 (2013), 404-413.  doi: 10.1007/s10958-013-1536-x.  Google Scholar

[22]

V. Rozko, Stability in terms of Lyapunov of discontinuous dynamic systems, Differ.Uravn., 11 (1975), 1005-1012.   Google Scholar

[23]

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equitations Singapore : World Scientific, 1995. Google Scholar

[24]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer, New York, 1988. Google Scholar

[25]

J. Valero, Finite and infinite-dimensional attractors of multivalued reaction-diffusion equations, Acta Mathematica Hungar., 88 (2000), 239-258.  doi: 10.1023/A:1006769315268.  Google Scholar

[26]

M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and N. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing Ⅲ. Long-Time Behaviour of Evolution Inclusions Solutions in Earth Data Analysis Springer, Heidelberg, 2012. Google Scholar

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