# American Institute of Mathematical Sciences

September  2015, 14(5): 1685-1704. doi: 10.3934/cpaa.2015.14.1685

## Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay

 1 Department of Mechanics and Mathematics, Karazin Kharkov National University, Kharkov, 61022 2 Department of Mechanics and Mathematics, V.N.Karazin Kharkiv National University, 4, Svobody Sqr., Kharkiv, 61077

Received  October 2014 Revised  February 2015 Published  June 2015

We deal with a class of parabolic nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in biology. We first prove well-posedness in a certain space of functions which are Lipschitz in time. This allows us to show that the model considered generates an evolution operator semigroup $S_t$ on a certain space of Lipschitz type functions over delay time interval. The operators $S_t$ are closed for all $t\ge 0$ and continuous for $t$ large enough. Our main result shows that the semigroup $S_t$ possesses compact global and exponential attractors of finite fractal dimension. Our argument is based on the recently developed method of quasi-stability estimates and involves some extension of the theory of global attractors for the case of closed evolutions.
Citation: Igor Chueshov, Alexander V. Rezounenko. Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1685-1704. doi: 10.3934/cpaa.2015.14.1685
##### References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations,, Amsterdam, (1992).   Google Scholar [2] R. Bellman and K. L. Cooke, Differential-difference Equations, in "Mathematics in Science and Engineering''. Vol. 6., New York-London: Academic Press, (1963).   Google Scholar [3] N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model,, \emph{SIAM. J. Appl. Math.}, 50 (1990), 1663.  doi: 10.1137/0150099.  Google Scholar [4] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta, (1999).   Google Scholar [5] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems,, Springer, (2015).   Google Scholar [6] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, \emph{J. of Dyn. and Diff. Equations}, 16 (2004), 469.  doi: 10.1007/s10884-004-4289-x.  Google Scholar [7] I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,, Mem. Amer. Math. Soc. 195 (2008), (2008).   Google Scholar [8] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-time Dynamics,, Springer Monographs in Mathematics. Springer, (2010).   Google Scholar [9] I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents,, in \emph{Nonlinear Hyperbolic PDEs, (2013), 1.   Google Scholar [10] 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,, \emph{Communications in Partial Differential Equations}, 39 (2014), 1965.  doi: 10.1080/03605302.2014.930484.  Google Scholar [11] I. Chueshov, I. Lasiecka and J. T. Webster, Flow-plate interactions: well-posedness and long-time behavior,, \emph{Discrete Continuous Dynamical Systems Ser. S}, 7 (2014), 925.  doi: 10.3934/dcdss.2014.7.925.  Google Scholar [12] I. Chueshov and A. Rezounenko, Dynamics of second order in time evolution equations with state-dependent delay,, \emph{Nonlinear Analysis: Theory, 123–-124 (2015), 126.  doi: 10.1016/j.na.2015.04.013.  Google Scholar [13] O. Diekmann, S. van Gils, S. Verduyn Lunel and H-O. Walther, Delay Equations: Functional, Complex, and Nonlinear Analysis,, Springer-Verlag, (1995).   Google Scholar [14] R. D. Driver, A two-body problem of classical electrodynamics: the one-dimensional case,, \emph{Ann. Physics}, 21 (1963), 122.  doi: 10.1016/0003-4916(63)90227-6.  Google Scholar [15] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations,, Research in Appl. Math. 37, (1994).   Google Scholar [16] W. E. Fitzgibbon, Semilinear functional differential equations in Banach space,, \emph{J. Differential Equations}, 29 (1978), 1.   Google Scholar [17] S. Gourley, J. So and J. Wu, Non-locality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics,, in \emph{Contemporary Mathematics, 124 (2004), 84.  doi: 10.1023/B:JOTH.0000047249.39572.6d.  Google Scholar [18] J. K. Hale, Theory of Functional Differential Equations,, Springer, (1977).   Google Scholar [19] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Amer. Math. Soc., (1988).   Google Scholar [20] F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications,, In \emph{Handbook of Differential Equations, (2006), 435.   Google Scholar [21] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, New York, (1981).   Google Scholar [22] T. Krisztin and O. Arino, The 2-dimensional attractor of a differential equation with state-dependent delay,, \emph{J. Dynamics and Differential Equations}, 13 (2001), 453.  doi: 10.1023/A:1016635223074.  Google Scholar [23] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Press, (1991).   Google Scholar [24] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969).   Google Scholar [25] J. L. Lions and E. Magenes, Problèmes aux Limites Non Homogénes et Applications,, Dunon, (1968).   Google Scholar [26] J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags,, \emph{Topol. Methods Nonlinear Anal.}, 3 (1994), 101.   Google Scholar [27] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, In \emph{Handbook of Differential Equations: Evolutionary Equations} (C. M. Dafermos and M. Pokorny eds.), (2008), 103.   Google Scholar [28] V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators,, \emph{Commun. Pure. Appl. Anal.}, 6 (2007), 481.  doi: 10.3934/cpaa.2007.6.481.  Google Scholar [29] A. V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays,, \emph{Journal of Mathematical Analysis and Applications}, 326 (2007), 1031.   Google Scholar [30] A. V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions,, \emph{Nonlinear Analysis: Theory, 70 (2009), 3978.  doi: 10.1016/j.na.2008.08.006.  Google Scholar [31] A. V. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space,, \emph{Nonlinear Analysis: Theory, 73 (2010), 1707.  doi: 10.1016/j.na.2010.05.005.  Google Scholar [32] A. V. Rezounenko, A condition on delay for differential equations with discrete state-dependent delay,, \emph{Journal of Mathematical Analysis and Applications}, 385 (2012), 506.  doi: 10.1016/j.jmaa.2011.06.070.  Google Scholar [33] A. V. Rezounenko and P. Zagalak, Non-local PDEs with discrete state-dependent delays: well-posedness in a metric space,, \emph{Discrete and Continuous Dynamical Systems - Series A}, 33 (2013), 819.  doi: 10.3934/dcds.2013.33.819.  Google Scholar [34] J. Simon, Compact sets in the space $L^p(0,T;B)$,, \emph{Annali di Mat. Pura ed Appl.}, 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar [35] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,, AMS, (1997).   Google Scholar [36] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988).   Google Scholar [37] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations,, \emph{Transactions of AMS}, 200 (1974), 395.   Google Scholar [38] H.-O. Walther, The solution manifold and $C^1$-smoothness for differential equations with state-dependent delay,, \emph{Journal of Differential Equations}, 195 (2003), 46.  doi: 10.1016/j.jde.2003.07.001.  Google Scholar [39] J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer-Verlag, (1996).   Google Scholar

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##### References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations,, Amsterdam, (1992).   Google Scholar [2] R. Bellman and K. L. Cooke, Differential-difference Equations, in "Mathematics in Science and Engineering''. Vol. 6., New York-London: Academic Press, (1963).   Google Scholar [3] N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model,, \emph{SIAM. J. Appl. Math.}, 50 (1990), 1663.  doi: 10.1137/0150099.  Google Scholar [4] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta, (1999).   Google Scholar [5] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems,, Springer, (2015).   Google Scholar [6] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, \emph{J. of Dyn. and Diff. Equations}, 16 (2004), 469.  doi: 10.1007/s10884-004-4289-x.  Google Scholar [7] I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,, Mem. Amer. Math. Soc. 195 (2008), (2008).   Google Scholar [8] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-time Dynamics,, Springer Monographs in Mathematics. Springer, (2010).   Google Scholar [9] I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents,, in \emph{Nonlinear Hyperbolic PDEs, (2013), 1.   Google Scholar [10] 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,, \emph{Communications in Partial Differential Equations}, 39 (2014), 1965.  doi: 10.1080/03605302.2014.930484.  Google Scholar [11] I. Chueshov, I. Lasiecka and J. T. Webster, Flow-plate interactions: well-posedness and long-time behavior,, \emph{Discrete Continuous Dynamical Systems Ser. S}, 7 (2014), 925.  doi: 10.3934/dcdss.2014.7.925.  Google Scholar [12] I. Chueshov and A. Rezounenko, Dynamics of second order in time evolution equations with state-dependent delay,, \emph{Nonlinear Analysis: Theory, 123–-124 (2015), 126.  doi: 10.1016/j.na.2015.04.013.  Google Scholar [13] O. Diekmann, S. van Gils, S. Verduyn Lunel and H-O. Walther, Delay Equations: Functional, Complex, and Nonlinear Analysis,, Springer-Verlag, (1995).   Google Scholar [14] R. D. Driver, A two-body problem of classical electrodynamics: the one-dimensional case,, \emph{Ann. Physics}, 21 (1963), 122.  doi: 10.1016/0003-4916(63)90227-6.  Google Scholar [15] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations,, Research in Appl. Math. 37, (1994).   Google Scholar [16] W. E. Fitzgibbon, Semilinear functional differential equations in Banach space,, \emph{J. Differential Equations}, 29 (1978), 1.   Google Scholar [17] S. Gourley, J. So and J. Wu, Non-locality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics,, in \emph{Contemporary Mathematics, 124 (2004), 84.  doi: 10.1023/B:JOTH.0000047249.39572.6d.  Google Scholar [18] J. K. Hale, Theory of Functional Differential Equations,, Springer, (1977).   Google Scholar [19] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Amer. Math. Soc., (1988).   Google Scholar [20] F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications,, In \emph{Handbook of Differential Equations, (2006), 435.   Google Scholar [21] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, New York, (1981).   Google Scholar [22] T. Krisztin and O. Arino, The 2-dimensional attractor of a differential equation with state-dependent delay,, \emph{J. Dynamics and Differential Equations}, 13 (2001), 453.  doi: 10.1023/A:1016635223074.  Google Scholar [23] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Press, (1991).   Google Scholar [24] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969).   Google Scholar [25] J. L. Lions and E. Magenes, Problèmes aux Limites Non Homogénes et Applications,, Dunon, (1968).   Google Scholar [26] J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags,, \emph{Topol. Methods Nonlinear Anal.}, 3 (1994), 101.   Google Scholar [27] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, In \emph{Handbook of Differential Equations: Evolutionary Equations} (C. M. Dafermos and M. Pokorny eds.), (2008), 103.   Google Scholar [28] V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators,, \emph{Commun. Pure. Appl. Anal.}, 6 (2007), 481.  doi: 10.3934/cpaa.2007.6.481.  Google Scholar [29] A. V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays,, \emph{Journal of Mathematical Analysis and Applications}, 326 (2007), 1031.   Google Scholar [30] A. V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions,, \emph{Nonlinear Analysis: Theory, 70 (2009), 3978.  doi: 10.1016/j.na.2008.08.006.  Google Scholar [31] A. V. Rezounenko, Non-linear partial differential equations with discrete state-dependent delays in a metric space,, \emph{Nonlinear Analysis: Theory, 73 (2010), 1707.  doi: 10.1016/j.na.2010.05.005.  Google Scholar [32] A. V. Rezounenko, A condition on delay for differential equations with discrete state-dependent delay,, \emph{Journal of Mathematical Analysis and Applications}, 385 (2012), 506.  doi: 10.1016/j.jmaa.2011.06.070.  Google Scholar [33] A. V. Rezounenko and P. Zagalak, Non-local PDEs with discrete state-dependent delays: well-posedness in a metric space,, \emph{Discrete and Continuous Dynamical Systems - Series A}, 33 (2013), 819.  doi: 10.3934/dcds.2013.33.819.  Google Scholar [34] J. Simon, Compact sets in the space $L^p(0,T;B)$,, \emph{Annali di Mat. Pura ed Appl.}, 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar [35] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,, AMS, (1997).   Google Scholar [36] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988).   Google Scholar [37] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations,, \emph{Transactions of AMS}, 200 (1974), 395.   Google Scholar [38] H.-O. Walther, The solution manifold and $C^1$-smoothness for differential equations with state-dependent delay,, \emph{Journal of Differential Equations}, 195 (2003), 46.  doi: 10.1016/j.jde.2003.07.001.  Google Scholar [39] J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer-Verlag, (1996).   Google Scholar
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