# American Institute of Mathematical Sciences

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.  Google Scholar [2] A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, Amsterdam, NorthHolland, 1992.  Google Scholar [3] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580, Springer-Verlag, Berlin, 1977.  Google Scholar [4] L. Boutet de Monvel, I. Chueshov and A. Rezounenko, Long-time behaviour of strong solutions of retarded nonlinear PDEs, Communications in Partial Differential Equations, 22 (1997), 1453-1474.   Google Scholar [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.  Google Scholar [6] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, English translation, 2002; http://www.emis.de/monographs/Chueshov/  Google Scholar [7] I. Chueshov, Monotone Random Systems: Theory and Applications, Lecture Notes Math. 1779, Springer, Berlin 2002.  Google Scholar [8] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer-Verlag, Berlin 2015.  Google Scholar [9] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dyn. Diff. Eqns., 16 (2004), 469-512.   Google Scholar [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.   Google Scholar [11] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Longtime Dynamics, Springer-Verlag, New York, 2010.  Google Scholar [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.  Google Scholar [13] I. Chueshov, I. 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.   Google Scholar [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).  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [18] M. Conti, E. M. Marchini and V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst., 27 (2010), 1535-1552.   Google Scholar [19] K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Applns, 86 (1982), 592-627.   Google Scholar [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.  Google Scholar [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.  Google Scholar [22] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Appl. Math. 37, Masson, Paris, 1994.  Google Scholar [23] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Cont. Dyn. Systems, 10 (2004), 211-238.   Google Scholar [24] W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differential Equations, 29 (1978), 1-14.   Google Scholar [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.   Google Scholar [26] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, Berlin, 1977.  Google Scholar [27] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.   Google Scholar [32] J. L. Lions, Quelques Méthodes de R´esolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.  Google Scholar [33] J. L. Lions, E. Magenes, Problèmes aux Limites Non Homogénes et Applications, Dunon, Paris, 1968.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] J. Mallet-Paret, R. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [45] A. P. S. Selvadurai, Elastic Analysis of Soil Foundation Interaction, Elsevier, Amsterdam, 1979. Google Scholar [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.  Google Scholar [47] R. E. Showalter, Monotone Operators in Banach space and Nonlinear Partial Differential Equations, AMS, Mathematical Surveys and Monographs, vol. 49,1997.  Google Scholar [48] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlaag, Berlin, 1988.  Google Scholar [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.  Google Scholar [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). Google Scholar [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.  Google Scholar [52] H.-O. Walther, On Poisson's state-dependent delay, Discrete Contin. Dyn. Syst., 33 (2013), 365-379.   Google Scholar [53] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.  Google Scholar

show all references

##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.  Google Scholar [2] A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, Amsterdam, NorthHolland, 1992.  Google Scholar [3] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580, Springer-Verlag, Berlin, 1977.  Google Scholar [4] L. Boutet de Monvel, I. Chueshov and A. Rezounenko, Long-time behaviour of strong solutions of retarded nonlinear PDEs, Communications in Partial Differential Equations, 22 (1997), 1453-1474.   Google Scholar [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.  Google Scholar [6] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, English translation, 2002; http://www.emis.de/monographs/Chueshov/  Google Scholar [7] I. Chueshov, Monotone Random Systems: Theory and Applications, Lecture Notes Math. 1779, Springer, Berlin 2002.  Google Scholar [8] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer-Verlag, Berlin 2015.  Google Scholar [9] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dyn. Diff. Eqns., 16 (2004), 469-512.   Google Scholar [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.   Google Scholar [11] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Longtime Dynamics, Springer-Verlag, New York, 2010.  Google Scholar [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.  Google Scholar [13] I. Chueshov, I. 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.   Google Scholar [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).  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [18] M. Conti, E. M. Marchini and V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst., 27 (2010), 1535-1552.   Google Scholar [19] K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Applns, 86 (1982), 592-627.   Google Scholar [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.  Google Scholar [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.  Google Scholar [22] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Appl. Math. 37, Masson, Paris, 1994.  Google Scholar [23] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Cont. Dyn. Systems, 10 (2004), 211-238.   Google Scholar [24] W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differential Equations, 29 (1978), 1-14.   Google Scholar [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.   Google Scholar [26] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, Berlin, 1977.  Google Scholar [27] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.   Google Scholar [32] J. L. Lions, Quelques Méthodes de R´esolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.  Google Scholar [33] J. L. Lions, E. Magenes, Problèmes aux Limites Non Homogénes et Applications, Dunon, Paris, 1968.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] J. Mallet-Paret, R. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [45] A. P. S. Selvadurai, Elastic Analysis of Soil Foundation Interaction, Elsevier, Amsterdam, 1979. Google Scholar [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.  Google Scholar [47] R. E. Showalter, Monotone Operators in Banach space and Nonlinear Partial Differential Equations, AMS, Mathematical Surveys and Monographs, vol. 49,1997.  Google Scholar [48] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlaag, Berlin, 1988.  Google Scholar [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.  Google Scholar [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). Google Scholar [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.  Google Scholar [52] H.-O. Walther, On Poisson's state-dependent delay, Discrete Contin. Dyn. Syst., 33 (2013), 365-379.   Google Scholar [53] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.  Google Scholar
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