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 and 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, North-Holland, 1992.

[2]

R. Bellman and K. L. Cooke, Differential-difference Equations, in "Mathematics in Science and Engineering''. Vol. 6. New York-London: Academic Press, XVI, 1963.

[3]

N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM. J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099.

[4]

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

[5]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015 (to be published).

[6]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. of Dyn. and Diff. Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x.

[7]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc. 195 (2008), no. 912, viii+183 pp.

[8]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-time Dynamics, Springer Monographs in Mathematics. Springer, New York, 2010.

[9]

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 Vol.6 (G. Alberti et al. eds.), AIMS, Springfield, 2013, 1-96.

[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, Communications in Partial Differential Equations, 39 (2014), 1965-1997. doi: 10.1080/03605302.2014.930484.

[11]

I. Chueshov, I. Lasiecka and J. T. Webster, Flow-plate interactions: well-posedness and long-time behavior, Discrete Continuous Dynamical Systems Ser. S, 7 (2014), 925-965. doi: 10.3934/dcdss.2014.7.925.

[12]

I. Chueshov and A. Rezounenko, Dynamics of second order in time evolution equations with state-dependent delay, Nonlinear Analysis: Theory, Methods and Applications, 123–-124 (2015), 126-149. doi: 10.1016/j.na.2015.04.013.

[13]

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

[14]

R. D. Driver, A two-body problem of classical electrodynamics: the one-dimensional case, Ann. Physics, 21 (1963), 122-142. doi: 10.1016/0003-4916(63)90227-6.

[15]

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

[16]

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

[17]

S. Gourley, J. So and J. Wu, Non-locality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics, in Contemporary Mathematics, Thematic Surveys (D. V. Anosov AND A. Skubachevskii eds.), Kluwer, Plenum, Dordrecht, NewYork, 2003, 84-120; (see also Journal of Mathematical Sciences, 124 (2004), 5119-5153). doi: 10.1023/B:JOTH.0000047249.39572.6d.

[18]

J. K. Hale, Theory of Functional Differential Equations, Springer, Berlin- Heidelberg- New York, 1977.

[19]

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

[20]

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

[21]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, New York, Springer, 1981.

[22]

T. Krisztin and O. Arino, The 2-dimensional attractor of a differential equation with state-dependent delay, J. Dynamics and Differential Equations, 13 (2001), 453-522. doi: 10.1023/A:1016635223074.

[23]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.

[24]

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

[25]

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

[26]

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.

[27]

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

[28]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure. Appl. Anal., 6 (2007), 481-486. doi: 10.3934/cpaa.2007.6.481.

[29]

A. V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays, Journal of Mathematical Analysis and Applications, 326 (2007), 1031-1045.

[30]

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.

[31]

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.

[32]

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.

[33]

A. V. Rezounenko and P. Zagalak, Non-local PDEs with discrete state-dependent delays: well-posedness in a metric space, Discrete and Continuous Dynamical Systems - Series A, 33 (2013), 819-835. doi: 10.3934/dcds.2013.33.819.

[34]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Mat. Pura ed Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[35]

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

[36]

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

[37]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Transactions of AMS, 200 (1974), 395-418.

[38]

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

[39]

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

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, Amsterdam, North-Holland, 1992.

[2]

R. Bellman and K. L. Cooke, Differential-difference Equations, in "Mathematics in Science and Engineering''. Vol. 6. New York-London: Academic Press, XVI, 1963.

[3]

N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM. J. Appl. Math., 50 (1990), 1663-1688. doi: 10.1137/0150099.

[4]

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

[5]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015 (to be published).

[6]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. of Dyn. and Diff. Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x.

[7]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc. 195 (2008), no. 912, viii+183 pp.

[8]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-time Dynamics, Springer Monographs in Mathematics. Springer, New York, 2010.

[9]

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 Vol.6 (G. Alberti et al. eds.), AIMS, Springfield, 2013, 1-96.

[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, Communications in Partial Differential Equations, 39 (2014), 1965-1997. doi: 10.1080/03605302.2014.930484.

[11]

I. Chueshov, I. Lasiecka and J. T. Webster, Flow-plate interactions: well-posedness and long-time behavior, Discrete Continuous Dynamical Systems Ser. S, 7 (2014), 925-965. doi: 10.3934/dcdss.2014.7.925.

[12]

I. Chueshov and A. Rezounenko, Dynamics of second order in time evolution equations with state-dependent delay, Nonlinear Analysis: Theory, Methods and Applications, 123–-124 (2015), 126-149. doi: 10.1016/j.na.2015.04.013.

[13]

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

[14]

R. D. Driver, A two-body problem of classical electrodynamics: the one-dimensional case, Ann. Physics, 21 (1963), 122-142. doi: 10.1016/0003-4916(63)90227-6.

[15]

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

[16]

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

[17]

S. Gourley, J. So and J. Wu, Non-locality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics, in Contemporary Mathematics, Thematic Surveys (D. V. Anosov AND A. Skubachevskii eds.), Kluwer, Plenum, Dordrecht, NewYork, 2003, 84-120; (see also Journal of Mathematical Sciences, 124 (2004), 5119-5153). doi: 10.1023/B:JOTH.0000047249.39572.6d.

[18]

J. K. Hale, Theory of Functional Differential Equations, Springer, Berlin- Heidelberg- New York, 1977.

[19]

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

[20]

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

[21]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, New York, Springer, 1981.

[22]

T. Krisztin and O. Arino, The 2-dimensional attractor of a differential equation with state-dependent delay, J. Dynamics and Differential Equations, 13 (2001), 453-522. doi: 10.1023/A:1016635223074.

[23]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.

[24]

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

[25]

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

[26]

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.

[27]

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

[28]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure. Appl. Anal., 6 (2007), 481-486. doi: 10.3934/cpaa.2007.6.481.

[29]

A. V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays, Journal of Mathematical Analysis and Applications, 326 (2007), 1031-1045.

[30]

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.

[31]

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.

[32]

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.

[33]

A. V. Rezounenko and P. Zagalak, Non-local PDEs with discrete state-dependent delays: well-posedness in a metric space, Discrete and Continuous Dynamical Systems - Series A, 33 (2013), 819-835. doi: 10.3934/dcds.2013.33.819.

[34]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Mat. Pura ed Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[35]

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

[36]

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

[37]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Transactions of AMS, 200 (1974), 395-418.

[38]

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

[39]

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

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