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

Global attractors of impulsive parabolic inclusions

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.

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.

[2]

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

[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.

[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.

[5]

E. M. BonottoM. C. BortolanA. 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.

[6]

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

[7]

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

[8]

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

[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.

[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.

[11]

G. IovaneO. 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.

[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.

[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.

[14]

O. V. KapustyanV. 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.

[15]

O. V. KapustyanP. 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.

[16]

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 and continuous dynamical systems, 34 (2014), 4155-4182. doi: 10.3934/dcds.2014.34.4155.

[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.

[18]

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

[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.

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[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.

[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.

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.

[2]

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

[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.

[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.

[5]

E. M. BonottoM. C. BortolanA. 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.

[6]

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

[7]

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

[8]

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

[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.

[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.

[11]

G. IovaneO. 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.

[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.

[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.

[14]

O. V. KapustyanV. 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.

[15]

O. V. KapustyanP. 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.

[16]

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 and continuous dynamical systems, 34 (2014), 4155-4182. doi: 10.3934/dcds.2014.34.4155.

[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.

[18]

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

[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.

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[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.

[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.

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