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The Penrose-Fife phase-field model with coupled dynamic boundary conditions
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 |
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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