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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

Sylvia Novo 1, , Carmen Núñez 1, , Rafael Obaya 2, and  Ana M. Sanz 3,

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.
Keywords: infinite delay., skew-product semiflows, Topological dynamics, finite delay, abstract functional differential equations, partial functional differential equations.
Mathematics Subject Classification: 37B55, 37C69, 37K57, 35R1.
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, 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.  Google Scholar

[2]

L. Arnold and I. D. Chueshov, Cooperative random and stochastic differential equations, Discrete Contin. Dyn. Syst., 7 (2001), 1-33.  Google Scholar

[3]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.  Google Scholar

[4]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcialaj Ekvac., 21 (1978), 11-41.  Google Scholar

[5]

H. R. Henríquez, Periodic solutions of quasi-linear partial functional differential equations with unbounded delay, Funkcialaj Ekvac., 37 (1994), 329-343.  Google Scholar

[6]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Math., 1473, Springer-Verlag, Berlin, Heidelberg, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

S. Ruan and J. Wu, Reaction-diffusion equations with infinite delay, Canad. Appl. Math. Quart., 2 (1994), 485-550.  Google Scholar

[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).  Google Scholar

[23]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

L. Arnold and I. D. Chueshov, Cooperative random and stochastic differential equations, Discrete Contin. Dyn. Syst., 7 (2001), 1-33.  Google Scholar

[3]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.  Google Scholar

[4]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcialaj Ekvac., 21 (1978), 11-41.  Google Scholar

[5]

H. R. Henríquez, Periodic solutions of quasi-linear partial functional differential equations with unbounded delay, Funkcialaj Ekvac., 37 (1994), 329-343.  Google Scholar

[6]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Math., 1473, Springer-Verlag, Berlin, Heidelberg, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

S. Ruan and J. Wu, Reaction-diffusion equations with infinite delay, Canad. Appl. Math. Quart., 2 (1994), 485-550.  Google Scholar

[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).  Google Scholar

[23]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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