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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 and Continuous Dynamical Systems, 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), 265-280. doi: 10.1080/02681119808806264. [2] L. Arnold and I. D. Chueshov, Cooperative random and stochastic differential equations, Discrete Contin. Dyn. Syst., 7 (2001), 1-33. [3] R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969. [4] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcialaj Ekvac., 21 (1978), 11-41. [5] H. R. Henríquez, Periodic solutions of quasi-linear partial functional differential equations with unbounded delay, Funkcialaj Ekvac., 37 (1994), 329-343. [6] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Math., 1473, Springer-Verlag, Berlin, Heidelberg, 1991. [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, Methods and Applications, 15, Taylor and Francis, London, 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), 21-55. 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, Springer, Berlin, 2003, 173-229. 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), 403-438. doi: 10.1007/BF01208484. [11] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications, 16, Birkhäuser, Basel, Boston, Berlin, 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), 1-44. 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), 1-35. [14] V. Muñoz-Villarragut, S. Novo and R. Obaya, Neutral functional differential equations with applications to compartmental systems, SIAM J. Math. Anal., 40 (2008), 1003-1028. 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), 589-619. 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, Lecture Notes in Math., 2065, Springer-Verlag, Berlin, Heidelberg, 2013, 185-264. 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), 623-646. 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), 1201-1231. 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), 1025-1053. 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), 279-303. doi: 10.1007/BF01053163. [21] S. Ruan and J. Wu, Reaction-diffusion equations with infinite delay, Canad. Appl. Math. Quart., 2 (1994), 485-550. [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), 320-358. [24] R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations, 113 (1994), 17-67. 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, 41, Amer. Math. Soc., Providence, RI, 1995. [27] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3. [28] J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, 119, Springer-Verlag, New York, 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), 265-280. doi: 10.1080/02681119808806264. [2] L. Arnold and I. D. Chueshov, Cooperative random and stochastic differential equations, Discrete Contin. Dyn. Syst., 7 (2001), 1-33. [3] R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969. [4] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcialaj Ekvac., 21 (1978), 11-41. [5] H. R. Henríquez, Periodic solutions of quasi-linear partial functional differential equations with unbounded delay, Funkcialaj Ekvac., 37 (1994), 329-343. [6] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Math., 1473, Springer-Verlag, Berlin, Heidelberg, 1991. [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, Methods and Applications, 15, Taylor and Francis, London, 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), 21-55. 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, Springer, Berlin, 2003, 173-229. 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), 403-438. doi: 10.1007/BF01208484. [11] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications, 16, Birkhäuser, Basel, Boston, Berlin, 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), 1-44. 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), 1-35. [14] V. Muñoz-Villarragut, S. Novo and R. Obaya, Neutral functional differential equations with applications to compartmental systems, SIAM J. Math. Anal., 40 (2008), 1003-1028. 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), 589-619. 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, Lecture Notes in Math., 2065, Springer-Verlag, Berlin, Heidelberg, 2013, 185-264. 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), 623-646. 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), 1201-1231. 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), 1025-1053. 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), 279-303. doi: 10.1007/BF01053163. [21] S. Ruan and J. Wu, Reaction-diffusion equations with infinite delay, Canad. Appl. Math. Quart., 2 (1994), 485-550. [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), 320-358. [24] R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations, 113 (1994), 17-67. 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, 41, Amer. Math. Soc., Providence, RI, 1995. [27] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3. [28] J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.
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