May 2018, 23(3): 991-1009. doi: 10.3934/dcdsb.2018139

Long term dynamics of second order-in-time stochastic evolution equations with state-dependent delay

1. 

Department of Mechanics and Mathematics, Kharkov National University, 61077, Kharkov, Ukraine

2. 

School of Mathematics & Statistics, Huazhong University of Science & Technology, Wuhan 430074, China

Dedicated to the memory of Igor Chueshov 1
1Died 23 April 2016

Received  September 2016 Revised  January 2017 Published  February 2018

Fund Project: Partially supported by the Chinese NSF grant no. 1157112 and NCET-12-0204, and the Spanish Ministerio de Economía y Competitividad project project MTM2015-63723-P

The well-posedness and asymptotic dynamics of second-order-in-time stochastic evolution equations with state-dependent delay is investigated. This class covers several important stochastic PDE models arising in the theory of nonlinear plates with additive noise. We first prove well-posedness in a certain space of functions which are $C^1$ in time. The solutions constructed generate a random dynamical system in a $C^1$-type space over the delay time interval. Our main result shows that this random dynamical system possesses compact global and exponential attractors of finite fractal dimension. To obtain this result we adapt the recently developed method of quasi-stability estimates to the random setting.

Citation: Igor Chueshov, Peter E. Kloeden, Meihua Yang. Long term dynamics of second order-in-time stochastic evolution equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 991-1009. doi: 10.3934/dcdsb.2018139
References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, Amsterdam, NorthHolland, 1992.

[3]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580, Springer-Verlag, Berlin, 1977.

[4]

L. Boutet de MonvelI. Chueshov and A. Rezounenko, Long-time behaviour of strong solutions of retarded nonlinear PDEs, Communications in Partial Differential Equations, 22 (1997), 1453-1474.

[5]

I. Chueshov, On a system of equations with delay that arises in aero-elasticity (Russian), Teor. Funktsii Funktsional. Anal. i Prilozhen., 54 (1990), 123-130; translation in J. Soviet Math., 58 (1992), 385-390.

[6]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, English translation, 2002; http://www.emis.de/monographs/Chueshov/

[7]

I. Chueshov, Monotone Random Systems: Theory and Applications, Lecture Notes Math. 1779, Springer, Berlin 2002.

[8]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer-Verlag, Berlin 2015.

[9]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dyn. Diff. Eqns., 16 (2004), 469-512.

[10]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs Amer. Math. Soc., 195 (2008), ⅷ+183 pp.

[11]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Longtime Dynamics, Springer-Verlag, New York, 2010.

[12]

I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents, In: Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations (HCDTE Lecture Notes, Part Ⅰ), AIMS on Applied Mathematics, G. Alberti et al. (Eds. ) AIMS, Springfield, 6 (2013), 1-96.

[13]

I. ChueshovI. Lasiecka and J. T. Webster, Attractors for delayed, non-rotational von Karman plates with applications to flow-structure interactions without any damping, Communications in Partial Differential Equations, 39 (2014), 1965-1997.

[14]

I. Chueshov and A. V. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations, C. R. Acad. Sci. Paris, Ser. I, 321 (1995), 607-612; (detailed version: Math. Physics, Analysis, Geometry, 2 (1995), 363-383).

[15]

I. Chueshov and A. Rezounenko, Dynamics of second order in time evolution equations with state-dependent delay, Nonlinear Analysis TMA, 123/124 (2015), 126-149.

[16]

I. Chueshov and A. Rezounenko, Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay, Communications on Pure and Applied Analysis, 14 (2015), 1685-1704.

[17]

I. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dyn. Diff. Eqns., 13 (2001), 355-380.

[18]

M. ContiE. M. Marchini and V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst., 27 (2010), 1535-1552.

[19]

K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Applns, 86 (1982), 592-627.

[20]

V. Danese, P. G. Geredeli and V. Pata, Exponential attractors for abstract equations with memory and applications to viscoelasticity, Discrete Contin. Dyn. Syst., 35 (2015), 2881-2904, arXiv: 1410.5051.

[21]

O. Diekmann, S. van Gils, S. Verduyn Lunel and H. -O. Walther, Delay Equations: Functional, Complex, and Nonlinear Analysis, Springer-Verlag, New York, 1995.

[22]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Appl. Math. 37, Masson, Paris, 1994.

[23]

P. FabrieC. GalusinskiA. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Cont. Dyn. Systems, 10 (2004), 211-238.

[24]

W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differential Equations, 29 (1978), 1-14.

[25]

M. J. Garrido-Atienza and J. Real, Existence and uniqueness of solutions for delay evolution equations of second order in time, J. Math. Anal. Appl., 283 (2003), 582-609.

[26]

J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, Berlin, 1977.

[27]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.

[28]

F. Hartung, T. Krisztin, H. -O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications. In: Canada, A., Drabek., P. and A. Fonda (Eds. ) Handbook of Differential Equations, Ordinary Differential Equations, vol. 3, Elsevier Science B. V., North Holland, 2006,435-545.

[29]

A. G. Kartsatos and L. P. Markov, An $L_2$-approach to second-order nonlinear functional evolutions involving m-accretive operators in Banach spaces, Differential Integral Equations, 14 (2001), 833-866.

[30]

T. Krisztin and O. Arino, The two-dimensional attractor of a differential equation with state-dependent delay, J. Dynam. Diff. Eqns., 13 (2001), 453-522. doi: 10.1023/A:1016635223074.

[31]

K. Kunisch and W. Schappacher, Necessary conditions for partial differential equations with delay to generate $C_0$-semigroups, J. Differential Equations, 50 (1983), 49-79.

[32]

J. L. Lions, Quelques Méthodes de R´esolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.

[33]

J. L. Lions, E. Magenes, Problèmes aux Limites Non Homogénes et Applications, Dunon, Paris, 1968.

[34]

J. Málek and J. Nečas, A finite dimensional attractor for three dimensional flow of incompressible fluids, J. Differential Equations, 127 (1996), 498-518. doi: 10.1006/jdeq.1996.0080.

[35]

J. Málek and D. Pražak, Large time behavior via the method of l-trajectories, J. Differential Equations, 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087.

[36]

J. Mallet-ParetR. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 3 (1994), 101-162. doi: 10.12775/TMNA.1994.006.

[37]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains. In: C. M. Dafermos, and M. Pokorny (Eds. ), Handbook of Differential Equations: Evolutionary Equations, Elsevier, Amsterdam, 4 (2008), 103-200.

[38]

V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure Appl. Anal., 9 (2010), 721-730. doi: 10.3934/cpaa.2010.9.721.

[39]

A. V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays, J. Math. Anal. Applns, 326 (2007), 1031-1045. doi: 10.1016/j.jmaa.2006.03.049.

[40]

A. V. Rezounenko, Differential equations with discrete state-dependent delay: Uniqueness and well-posedness in the space of continuous functions, Nonlinear Analysis: Theory, Methods and Applications, 70 (2009), 3978-3986. doi: 10.1016/j.na.2008.08.006.

[41]

A. V. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space, Nonlinear Analysis: Theory, Methods and Applications, 73 (2010), 1707-1714. doi: 10.1016/j.na.2010.05.005.

[42]

A. V. Rezounenko, A condition on delay for differential equations with discrete state-dependent delay, Journal of Mathematical Analysis and Applications, 385 (2012), 506-516. doi: 10.1016/j.jmaa.2011.06.070.

[43]

A. V. Rezounenko and P. Zagalak, Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space, Discrete Cont. Dyn.l Systems, 33 (2013), 819-835.

[44]

W. M. Ruess, Existence of solutions to partial differential equations with delay. In: Theory and Applications of Nonlinear Operators of Accretive Monotone type, Lecture Notes Pure Appl. Math., 178 (1996), 259-288.

[45]

A. P. S. Selvadurai, Elastic Analysis of Soil Foundation Interaction, Elsevier, Amsterdam, 1979.

[46]

A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch PDE: Anal Comp, 1 (2013), 241-281. doi: 10.1007/s40072-013-0007-1.

[47]

R. E. Showalter, Monotone Operators in Banach space and Nonlinear Partial Differential Equations, AMS, Mathematical Surveys and Monographs, vol. 49,1997.

[48]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlaag, Berlin, 1988.

[49]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Transactions of AMS, 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3.

[50]

V. Z. Vlasov and U. N. Leontiev, Beams, Plates, and Shells on Elastic Foundation, Israel Program for Scientific Translations, Jerusalem, 1966 (translated from Russian).

[51]

H.-O. Walther, The solution manifold and $C^1$-smoothness for differential equations with state-dependent delay, J. Differential Equations, 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.

[52]

H.-O. Walther, On Poisson's state-dependent delay, Discrete Contin. Dyn. Syst., 33 (2013), 365-379.

[53]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, Amsterdam, NorthHolland, 1992.

[3]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580, Springer-Verlag, Berlin, 1977.

[4]

L. Boutet de MonvelI. Chueshov and A. Rezounenko, Long-time behaviour of strong solutions of retarded nonlinear PDEs, Communications in Partial Differential Equations, 22 (1997), 1453-1474.

[5]

I. Chueshov, On a system of equations with delay that arises in aero-elasticity (Russian), Teor. Funktsii Funktsional. Anal. i Prilozhen., 54 (1990), 123-130; translation in J. Soviet Math., 58 (1992), 385-390.

[6]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, English translation, 2002; http://www.emis.de/monographs/Chueshov/

[7]

I. Chueshov, Monotone Random Systems: Theory and Applications, Lecture Notes Math. 1779, Springer, Berlin 2002.

[8]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer-Verlag, Berlin 2015.

[9]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dyn. Diff. Eqns., 16 (2004), 469-512.

[10]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs Amer. Math. Soc., 195 (2008), ⅷ+183 pp.

[11]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Longtime Dynamics, Springer-Verlag, New York, 2010.

[12]

I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents, In: Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations (HCDTE Lecture Notes, Part Ⅰ), AIMS on Applied Mathematics, G. Alberti et al. (Eds. ) AIMS, Springfield, 6 (2013), 1-96.

[13]

I. ChueshovI. Lasiecka and J. T. Webster, Attractors for delayed, non-rotational von Karman plates with applications to flow-structure interactions without any damping, Communications in Partial Differential Equations, 39 (2014), 1965-1997.

[14]

I. Chueshov and A. V. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations, C. R. Acad. Sci. Paris, Ser. I, 321 (1995), 607-612; (detailed version: Math. Physics, Analysis, Geometry, 2 (1995), 363-383).

[15]

I. Chueshov and A. Rezounenko, Dynamics of second order in time evolution equations with state-dependent delay, Nonlinear Analysis TMA, 123/124 (2015), 126-149.

[16]

I. Chueshov and A. Rezounenko, Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay, Communications on Pure and Applied Analysis, 14 (2015), 1685-1704.

[17]

I. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dyn. Diff. Eqns., 13 (2001), 355-380.

[18]

M. ContiE. M. Marchini and V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst., 27 (2010), 1535-1552.

[19]

K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Applns, 86 (1982), 592-627.

[20]

V. Danese, P. G. Geredeli and V. Pata, Exponential attractors for abstract equations with memory and applications to viscoelasticity, Discrete Contin. Dyn. Syst., 35 (2015), 2881-2904, arXiv: 1410.5051.

[21]

O. Diekmann, S. van Gils, S. Verduyn Lunel and H. -O. Walther, Delay Equations: Functional, Complex, and Nonlinear Analysis, Springer-Verlag, New York, 1995.

[22]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Appl. Math. 37, Masson, Paris, 1994.

[23]

P. FabrieC. GalusinskiA. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Cont. Dyn. Systems, 10 (2004), 211-238.

[24]

W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differential Equations, 29 (1978), 1-14.

[25]

M. J. Garrido-Atienza and J. Real, Existence and uniqueness of solutions for delay evolution equations of second order in time, J. Math. Anal. Appl., 283 (2003), 582-609.

[26]

J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, Berlin, 1977.

[27]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.

[28]

F. Hartung, T. Krisztin, H. -O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications. In: Canada, A., Drabek., P. and A. Fonda (Eds. ) Handbook of Differential Equations, Ordinary Differential Equations, vol. 3, Elsevier Science B. V., North Holland, 2006,435-545.

[29]

A. G. Kartsatos and L. P. Markov, An $L_2$-approach to second-order nonlinear functional evolutions involving m-accretive operators in Banach spaces, Differential Integral Equations, 14 (2001), 833-866.

[30]

T. Krisztin and O. Arino, The two-dimensional attractor of a differential equation with state-dependent delay, J. Dynam. Diff. Eqns., 13 (2001), 453-522. doi: 10.1023/A:1016635223074.

[31]

K. Kunisch and W. Schappacher, Necessary conditions for partial differential equations with delay to generate $C_0$-semigroups, J. Differential Equations, 50 (1983), 49-79.

[32]

J. L. Lions, Quelques Méthodes de R´esolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.

[33]

J. L. Lions, E. Magenes, Problèmes aux Limites Non Homogénes et Applications, Dunon, Paris, 1968.

[34]

J. Málek and J. Nečas, A finite dimensional attractor for three dimensional flow of incompressible fluids, J. Differential Equations, 127 (1996), 498-518. doi: 10.1006/jdeq.1996.0080.

[35]

J. Málek and D. Pražak, Large time behavior via the method of l-trajectories, J. Differential Equations, 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087.

[36]

J. Mallet-ParetR. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 3 (1994), 101-162. doi: 10.12775/TMNA.1994.006.

[37]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains. In: C. M. Dafermos, and M. Pokorny (Eds. ), Handbook of Differential Equations: Evolutionary Equations, Elsevier, Amsterdam, 4 (2008), 103-200.

[38]

V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure Appl. Anal., 9 (2010), 721-730. doi: 10.3934/cpaa.2010.9.721.

[39]

A. V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays, J. Math. Anal. Applns, 326 (2007), 1031-1045. doi: 10.1016/j.jmaa.2006.03.049.

[40]

A. V. Rezounenko, Differential equations with discrete state-dependent delay: Uniqueness and well-posedness in the space of continuous functions, Nonlinear Analysis: Theory, Methods and Applications, 70 (2009), 3978-3986. doi: 10.1016/j.na.2008.08.006.

[41]

A. V. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space, Nonlinear Analysis: Theory, Methods and Applications, 73 (2010), 1707-1714. doi: 10.1016/j.na.2010.05.005.

[42]

A. V. Rezounenko, A condition on delay for differential equations with discrete state-dependent delay, Journal of Mathematical Analysis and Applications, 385 (2012), 506-516. doi: 10.1016/j.jmaa.2011.06.070.

[43]

A. V. Rezounenko and P. Zagalak, Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space, Discrete Cont. Dyn.l Systems, 33 (2013), 819-835.

[44]

W. M. Ruess, Existence of solutions to partial differential equations with delay. In: Theory and Applications of Nonlinear Operators of Accretive Monotone type, Lecture Notes Pure Appl. Math., 178 (1996), 259-288.

[45]

A. P. S. Selvadurai, Elastic Analysis of Soil Foundation Interaction, Elsevier, Amsterdam, 1979.

[46]

A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch PDE: Anal Comp, 1 (2013), 241-281. doi: 10.1007/s40072-013-0007-1.

[47]

R. E. Showalter, Monotone Operators in Banach space and Nonlinear Partial Differential Equations, AMS, Mathematical Surveys and Monographs, vol. 49,1997.

[48]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlaag, Berlin, 1988.

[49]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Transactions of AMS, 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3.

[50]

V. Z. Vlasov and U. N. Leontiev, Beams, Plates, and Shells on Elastic Foundation, Israel Program for Scientific Translations, Jerusalem, 1966 (translated from Russian).

[51]

H.-O. Walther, The solution manifold and $C^1$-smoothness for differential equations with state-dependent delay, J. Differential Equations, 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.

[52]

H.-O. Walther, On Poisson's state-dependent delay, Discrete Contin. Dyn. Syst., 33 (2013), 365-379.

[53]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.

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