# American Institute of Mathematical Sciences

• Previous Article
The Penrose-Fife phase-field model with coupled dynamic boundary conditions
• DCDS Home
• This Issue
• Next Article
The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain
October  2014, 34(10): 4291-4321. doi: 10.3934/dcds.2014.34.4291

## Skew-product semiflows for non-autonomous partial functional differential equations with delay

 1 Departamento de Matemática Aplicada, E. Ingenierías Industriales, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid 2 Departamento de Matemática Aplicada, Escuela de Ingenierías Industriales and member of IMUVA, Instituto de Matemáticas, Universidad de Valladolid, 47011 Valladolid 3 Departamento de Didáctica de las Ciencias Experimentales, Sociales y de la Matemática, E.U. Educación, Universidad de Valladolid, 34004 Palencia, Spain

Received  January 2013 Revised  March 2013 Published  April 2014

A detailed dynamical study of the skew-product semiflows induced by families of AFDEs with infinite delay on a Banach space is carried over. Applications are given for families of non-autonomous quasimonotone reaction-diffusion PFDEs with delay in the nonlinear reaction terms, both with finite and infinite delay. In this monotone setting, relations among the classical concepts of sub and super solutions and the dynamical concept of semi-equilibria are established, and some results on the existence of minimal semiflows with a particular dynamical structure are derived.
Citation: Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291
##### References:
 [1] L. Arnold and I. D. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications,, Dynam. Stability Systems, 13 (1998), 265. doi: 10.1080/02681119808806264. [2] L. Arnold and I. D. Chueshov, Cooperative random and stochastic differential equations,, Discrete Contin. Dyn. Syst., 7 (2001), 1. [3] R. Ellis, Lectures on Topological Dynamics,, Benjamin, (1969). [4] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcialaj Ekvac., 21 (1978), 11. [5] H. R. Henríquez, Periodic solutions of quasi-linear partial functional differential equations with unbounded delay,, Funkcialaj Ekvac., 37 (1994), 329. [6] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay,, Lecture Notes in Math., (1473). [7] Y. Hino, T. Naito, N. V. Minh and J. S. Shin, Almost Periodic Solutions of Differential Equations in Banach Spaces,, Stability and Control: Theory, (2002). [8] J. Jiang and X.-Q. Zhao, Convergence in monotone and uniformly stable skew-product semiflows with applications,, J. Reine Angew. Math., 589 (2005), 21. doi: 10.1515/crll.2005.2005.589.21. [9] R. Johnson and F. Mantellini, Non-autonomous differential equations,, in Dynamical Systems Lecture Notes in Math., (1822), 173. doi: 10.1007/978-3-540-45204-1_3. [10] R. Johnson and J. Moser, The rotation number for almost periodic differential equations,, Commun. Math. Phys., 84 (1982), 403. doi: 10.1007/BF01208484. [11] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Progress in Nonlinear Differential Equations and Their Applications, (1995). doi: 10.1007/978-3-0348-0557-5. [12] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1. doi: 10.2307/2001590. [13] R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence,, J. Reine Angew. Math., 413 (1991), 1. [14] V. Muñoz-Villarragut, S. Novo and R. Obaya, Neutral functional differential equations with applications to compartmental systems,, SIAM J. Math. Anal., 40 (2008), 1003. doi: 10.1137/070711177. [15] S. Novo, C. Núñez and R. Obaya, Almost automorphic and almost periodic dynamics for quasimonotone non-autonomous functional differential equations,, J. Dynamics Differential Equations, 17 (2005), 589. doi: 10.1007/s10884-005-5814-2. [16] S. Novo and R. Obaya, Non-autonomous functional differential equations and applications,, in Stability and Bifurcation for Non-Autonomous Differential Equations, (2065), 185. doi: 10.1007/978-3-642-32906-7_4. [17] S. Novo, R. Obaya and A. M. Sanz, Stability and extensibility results for abstract skew-product semiflows,, J. Differential Equations, 235 (2007), 623. doi: 10.1016/j.jde.2006.12.009. [18] S. Novo, R. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications,, J. Dynam. Differential Equations, 25 (2013), 1201. doi: 10.1007/s10884-013-9337-y. [19] S. Novo, R. Obaya and V. M. Villarragut, Exponential ordering for nonautonomous neutral functional differential equations,, SIAM J. Math. Anal., 41 (2009), 1025. doi: 10.1137/080744682. [20] P. Poláčik and I. Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations,, J. Dynamics Differential Equations, 5 (1993), 279. doi: 10.1007/BF01053163. [21] S. Ruan and J. Wu, Reaction-diffusion equations with infinite delay,, Canad. Appl. Math. Quart., 2 (1994), 485. [22] R. J. Sacker and G. R. Sell, Lifting properties in skew-products flows with applications to differential equations,, Mem. Amer. Math. Soc., 11 (1977). [23] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems,, J. Differential Equations, 27 (1978), 320. [24] R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces,, J. Differential Equations, 113 (1994), 17. doi: 10.1006/jdeq.1994.1113. [25] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows,, Mem. Amer. Math. Soc., 136 (1998). doi: 10.1090/memo/0647. [26] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995). [27] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations,, Trans. Amer. Math. Soc., 200 (1974), 395. doi: 10.1090/S0002-9947-1974-0382808-3. [28] J. Wu, Theory and Applications of Partial Functional Differential Equations,, Applied Mathematical Sciences, (1996). doi: 10.1007/978-1-4612-4050-1.

show all references

##### References:
 [1] L. Arnold and I. D. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications,, Dynam. Stability Systems, 13 (1998), 265. doi: 10.1080/02681119808806264. [2] L. Arnold and I. D. Chueshov, Cooperative random and stochastic differential equations,, Discrete Contin. Dyn. Syst., 7 (2001), 1. [3] R. Ellis, Lectures on Topological Dynamics,, Benjamin, (1969). [4] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcialaj Ekvac., 21 (1978), 11. [5] H. R. Henríquez, Periodic solutions of quasi-linear partial functional differential equations with unbounded delay,, Funkcialaj Ekvac., 37 (1994), 329. [6] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay,, Lecture Notes in Math., (1473). [7] Y. Hino, T. Naito, N. V. Minh and J. S. Shin, Almost Periodic Solutions of Differential Equations in Banach Spaces,, Stability and Control: Theory, (2002). [8] J. Jiang and X.-Q. Zhao, Convergence in monotone and uniformly stable skew-product semiflows with applications,, J. Reine Angew. Math., 589 (2005), 21. doi: 10.1515/crll.2005.2005.589.21. [9] R. Johnson and F. Mantellini, Non-autonomous differential equations,, in Dynamical Systems Lecture Notes in Math., (1822), 173. doi: 10.1007/978-3-540-45204-1_3. [10] R. Johnson and J. Moser, The rotation number for almost periodic differential equations,, Commun. Math. Phys., 84 (1982), 403. doi: 10.1007/BF01208484. [11] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Progress in Nonlinear Differential Equations and Their Applications, (1995). doi: 10.1007/978-3-0348-0557-5. [12] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1. doi: 10.2307/2001590. [13] R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence,, J. Reine Angew. Math., 413 (1991), 1. [14] V. Muñoz-Villarragut, S. Novo and R. Obaya, Neutral functional differential equations with applications to compartmental systems,, SIAM J. Math. Anal., 40 (2008), 1003. doi: 10.1137/070711177. [15] S. Novo, C. Núñez and R. Obaya, Almost automorphic and almost periodic dynamics for quasimonotone non-autonomous functional differential equations,, J. Dynamics Differential Equations, 17 (2005), 589. doi: 10.1007/s10884-005-5814-2. [16] S. Novo and R. Obaya, Non-autonomous functional differential equations and applications,, in Stability and Bifurcation for Non-Autonomous Differential Equations, (2065), 185. doi: 10.1007/978-3-642-32906-7_4. [17] S. Novo, R. Obaya and A. M. Sanz, Stability and extensibility results for abstract skew-product semiflows,, J. Differential Equations, 235 (2007), 623. doi: 10.1016/j.jde.2006.12.009. [18] S. Novo, R. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications,, J. Dynam. Differential Equations, 25 (2013), 1201. doi: 10.1007/s10884-013-9337-y. [19] S. Novo, R. Obaya and V. M. Villarragut, Exponential ordering for nonautonomous neutral functional differential equations,, SIAM J. Math. Anal., 41 (2009), 1025. doi: 10.1137/080744682. [20] P. Poláčik and I. Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations,, J. Dynamics Differential Equations, 5 (1993), 279. doi: 10.1007/BF01053163. [21] S. Ruan and J. Wu, Reaction-diffusion equations with infinite delay,, Canad. Appl. Math. Quart., 2 (1994), 485. [22] R. J. Sacker and G. R. Sell, Lifting properties in skew-products flows with applications to differential equations,, Mem. Amer. Math. Soc., 11 (1977). [23] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems,, J. Differential Equations, 27 (1978), 320. [24] R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces,, J. Differential Equations, 113 (1994), 17. doi: 10.1006/jdeq.1994.1113. [25] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows,, Mem. Amer. Math. Soc., 136 (1998). doi: 10.1090/memo/0647. [26] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995). [27] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations,, Trans. Amer. Math. Soc., 200 (1974), 395. doi: 10.1090/S0002-9947-1974-0382808-3. [28] J. Wu, Theory and Applications of Partial Functional Differential Equations,, Applied Mathematical Sciences, (1996). doi: 10.1007/978-1-4612-4050-1.
 [1] Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115 [2] Rafael Obaya, Ana M. Sanz. Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-24. doi: 10.3934/dcdsb.2018338 [3] Fuke Wu, Shigeng Hu. The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1065-1094. doi: 10.3934/dcds.2012.32.1065 [4] Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687 [5] Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167 [6] Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 [7] Klaudiusz Wójcik, Piotr Zgliczyński. Topological horseshoes and delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 827-852. doi: 10.3934/dcds.2005.12.827 [8] Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031 [9] Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167 [10] Tomás Caraballo, P.E. Kloeden, Pedro Marín-Rubio. Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 177-196. doi: 10.3934/dcds.2007.19.177 [11] Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038 [12] Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293 [13] Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793 [14] Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633 [15] Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295 [16] Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351 [17] Jiaohui Xu, Tomás Caraballo. Long time behavior of fractional impulsive stochastic differential equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2719-2743. doi: 10.3934/dcdsb.2018272 [18] Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, José A. Langa. Equi-attraction and continuity of attractors for skew-product semiflows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2949-2967. doi: 10.3934/dcdsb.2016081 [19] Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure & Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016 [20] Olesya V. Solonukha. On nonlinear and quasiliniear elliptic functional differential equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 869-893. doi: 10.3934/dcdss.2016033

2017 Impact Factor: 1.179