• Previous Article
    The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain
  • DCDS Home
  • This Issue
  • Next Article
    The Penrose-Fife phase-field model with coupled dynamic boundary conditions
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.  Google Scholar

[2]

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

[3]

R. Ellis, Lectures on Topological Dynamics,, Benjamin, (1969).   Google Scholar

[4]

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

[5]

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

[6]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay,, Lecture Notes in Math., (1473).   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, (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.  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), 173.  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.  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, (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.  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.   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.  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.  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, (2065), 185.  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.  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.  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.  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.  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.   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.   Google Scholar

[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.  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, (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.  doi: 10.1090/S0002-9947-1974-0382808-3.  Google Scholar

[28]

J. Wu, Theory and Applications of Partial Functional Differential Equations,, Applied Mathematical Sciences, (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.  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.   Google Scholar

[3]

R. Ellis, Lectures on Topological Dynamics,, Benjamin, (1969).   Google Scholar

[4]

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

[5]

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

[6]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay,, Lecture Notes in Math., (1473).   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, (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.  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), 173.  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.  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, (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.  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.   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.  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.  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, (2065), 185.  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.  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.  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.  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.  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.   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.   Google Scholar

[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.  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, (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.  doi: 10.1090/S0002-9947-1974-0382808-3.  Google Scholar

[28]

J. Wu, Theory and Applications of Partial Functional Differential Equations,, Applied Mathematical Sciences, (1996).  doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[1]

Stefan Ruschel, Serhiy Yanchuk. The Spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[2]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[3]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[4]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[5]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[6]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[7]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[8]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

[9]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[10]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116

[11]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[12]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268

[13]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[14]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[15]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305

[16]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[17]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[18]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[19]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[20]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (3)

[Back to Top]