November  2013, 33(11&12): 4795-4810. doi: 10.3934/dcds.2013.33.4795

Floquet representations and asymptotic behavior of periodic evolution families

1. 

Universität Tübingen, Mathematisch-Naturwissenschaftliche Fakultät, Auf der Morgenstelle 10, D-72076 Tübingen, Germany, Germany, Germany

Received  August 2011 Revised  September 2011 Published  May 2013

We use semigroup techniques to describe the asymptotic behavior of contractive, periodic evolution families on Hilbert spaces. In particular, we show that such evolution families converge almost weakly to a Floquet representation with discrete spectrum.
Citation: Fatih Bayazit, Ulrich Groh, Rainer Nagel. Floquet representations and asymptotic behavior of periodic evolution families. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4795-4810. doi: 10.3934/dcds.2013.33.4795
References:
[1]

H. Amann, "Gewöhnliche Differentialgleichungen," de Gruyter, 1983.

[2]

H. Amann, "Ordinary Differential Equations: An Introduction to Nonlinear Analysis," de Gruyter, 1990. doi: 10.1515/9783110853698.

[3]

H. Bercovici, C. Foiaş, L. Kèrchy and B. Nagy, "Harmonic Analysis of Operators on Hilbert Space," Springer-Verlag, 2010.

[4]

C. Chicone, "Ordinary Differential Equations with Applications," Springer-Verlag, 2006.

[5]

C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," American Mathematical Society, 1999.

[6]

J. Conway, "A Course in Functional Analysis," Springer-Verlag, 1997.

[7]

D. Daners and P. Koch Medina, "Abstract Evolution Equations, Periodic Problems and Applications," Pitman Research Notes, 1992.

[8]

N. Dunford and J. Schwartz, "Linear Operators Part I: General Theory," Interscience, 1958.

[9]

T. Eisner, Embedding operators into strongly continuous semigroups, Archiv Math., 92 (2009), 451-460. doi: 10.1007/s00013-009-3154-x.

[10]

T. Eisner, "Stability of Operators and $C_0$-Semigroups," Birkhäuser-Verlag, 2010.

[11]

K.-J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations," Springer-Verlag, 2000.

[12]

G. Floquet, "Sur la Théorie des Équations Différentielles Linèaires," Gauthier-Villars, 1879.

[13]

G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. Ècole Norm. Sup., 12 (1883), 47-88.

[14]

J. A. Goldstein, "Semigroups of Operators and Applications," Oxford University Press, 1985.

[15]

M. Haase, Spectral properties of operator logarithms, Math. Z., 245 (2003), 761-779. doi: 10.1007/s00209-003-0569-0.

[16]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Springer-Verlag, 1981.

[17]

H. Heuser, "Funktionalanalysis - Theorie und Anwendungen," Teubner-Verlag, 2006.

[18]

H. Junghenn, Tensor products and almost periodicity, Proc. Amer. Math. Soc., 43 (1974), 99-105. doi: 10.1090/S0002-9939-1974-0365223-3.

[19]

R. Nagel, Semigroup methods for nonautonomous Cauchy problems, Evolution Equations, Lecture Notes in Pure and Appl. Math., 168 (1995), 301-316.

[20]

H. H. Schaefer, "Banach Lattices and Positive Operators," Springer-Verlag, 1974.

[21]

R. Schnaubelt, Well-posedness and asymptotic behaviour of non-autonomous linear evolution equations, Prog. Nonlinear Differential Equations Appl., 50 (2002), 311-338.

[22]

A. Stokes, A Floquet theory for functional differential equation, Proc. Nat. Acad. Sciences USA, 48 (1962), 1330-1334. doi: 10.1073/pnas.48.8.1330.

show all references

References:
[1]

H. Amann, "Gewöhnliche Differentialgleichungen," de Gruyter, 1983.

[2]

H. Amann, "Ordinary Differential Equations: An Introduction to Nonlinear Analysis," de Gruyter, 1990. doi: 10.1515/9783110853698.

[3]

H. Bercovici, C. Foiaş, L. Kèrchy and B. Nagy, "Harmonic Analysis of Operators on Hilbert Space," Springer-Verlag, 2010.

[4]

C. Chicone, "Ordinary Differential Equations with Applications," Springer-Verlag, 2006.

[5]

C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," American Mathematical Society, 1999.

[6]

J. Conway, "A Course in Functional Analysis," Springer-Verlag, 1997.

[7]

D. Daners and P. Koch Medina, "Abstract Evolution Equations, Periodic Problems and Applications," Pitman Research Notes, 1992.

[8]

N. Dunford and J. Schwartz, "Linear Operators Part I: General Theory," Interscience, 1958.

[9]

T. Eisner, Embedding operators into strongly continuous semigroups, Archiv Math., 92 (2009), 451-460. doi: 10.1007/s00013-009-3154-x.

[10]

T. Eisner, "Stability of Operators and $C_0$-Semigroups," Birkhäuser-Verlag, 2010.

[11]

K.-J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations," Springer-Verlag, 2000.

[12]

G. Floquet, "Sur la Théorie des Équations Différentielles Linèaires," Gauthier-Villars, 1879.

[13]

G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. Ècole Norm. Sup., 12 (1883), 47-88.

[14]

J. A. Goldstein, "Semigroups of Operators and Applications," Oxford University Press, 1985.

[15]

M. Haase, Spectral properties of operator logarithms, Math. Z., 245 (2003), 761-779. doi: 10.1007/s00209-003-0569-0.

[16]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Springer-Verlag, 1981.

[17]

H. Heuser, "Funktionalanalysis - Theorie und Anwendungen," Teubner-Verlag, 2006.

[18]

H. Junghenn, Tensor products and almost periodicity, Proc. Amer. Math. Soc., 43 (1974), 99-105. doi: 10.1090/S0002-9939-1974-0365223-3.

[19]

R. Nagel, Semigroup methods for nonautonomous Cauchy problems, Evolution Equations, Lecture Notes in Pure and Appl. Math., 168 (1995), 301-316.

[20]

H. H. Schaefer, "Banach Lattices and Positive Operators," Springer-Verlag, 1974.

[21]

R. Schnaubelt, Well-posedness and asymptotic behaviour of non-autonomous linear evolution equations, Prog. Nonlinear Differential Equations Appl., 50 (2002), 311-338.

[22]

A. Stokes, A Floquet theory for functional differential equation, Proc. Nat. Acad. Sciences USA, 48 (1962), 1330-1334. doi: 10.1073/pnas.48.8.1330.

[1]

Jin Liang, James H. Liu, Ti-Jun Xiao. Nonlocal Cauchy problems for nonautonomous evolution equations. Communications on Pure and Applied Analysis, 2006, 5 (3) : 529-535. doi: 10.3934/cpaa.2006.5.529

[2]

Nobuyuki Kato. Linearized stability and asymptotic properties for abstract boundary value functional evolution problems. Conference Publications, 1998, 1998 (Special) : 371-387. doi: 10.3934/proc.1998.1998.371

[3]

Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171

[4]

Xuewei Ju, Desheng Li. Global synchronising behavior of evolution equations with exponentially growing nonautonomous forcing. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1921-1944. doi: 10.3934/cpaa.2018091

[5]

Davor Dragičević. Admissibility and polynomial dichotomies for evolution families. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1321-1336. doi: 10.3934/cpaa.2020064

[6]

Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations and Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032

[7]

Mihail Megan, Adina Luminiţa Sasu, Bogdan Sasu. Discrete admissibility and exponential dichotomy for evolution families. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 383-397. doi: 10.3934/dcds.2003.9.383

[8]

Christian Pötzsche, Evamaria Russ. Topological decoupling and linearization of nonautonomous evolution equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1235-1268. doi: 10.3934/dcdss.2016050

[9]

N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711

[10]

Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure and Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475

[11]

Zhe Xie, Jiangwei Zhang, Yongqin Xie. Asymptotic behavior of quasi-linear evolution equations on time-dependent product spaces. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022171

[12]

Antonio Segatti. Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 801-820. doi: 10.3934/dcds.2006.14.801

[13]

Chiu-Yen Kao, Yuan Lou, Wenxian Shen. Evolution of mixed dispersal in periodic environments. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2047-2072. doi: 10.3934/dcdsb.2012.17.2047

[14]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Periodic solutions for implicit evolution inclusions. Evolution Equations and Control Theory, 2019, 8 (3) : 621-631. doi: 10.3934/eect.2019029

[15]

Angela A. Albanese, Elisabetta M. Mangino. Analytic semigroups and some degenerate evolution equations defined on domains with corners. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 595-615. doi: 10.3934/dcds.2015.35.595

[16]

Noboru Okazawa, Kentarou Yoshii. Linear evolution equations with strongly measurable families and application to the Dirac equation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 723-744. doi: 10.3934/dcdss.2011.4.723

[17]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2725-2737. doi: 10.3934/dcds.2020383

[18]

Pengyu Chen, Xuping Zhang, Yongxiang Li. A blowup alternative result for fractional nonautonomous evolution equation of Volterra type. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1975-1992. doi: 10.3934/cpaa.2018094

[19]

Pierre Fabrie, Alain Miranville. Exponential attractors for nonautonomous first-order evolution equations. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 225-240. doi: 10.3934/dcds.1998.4.225

[20]

Norbert Koksch, Stefan Siegmund. Feedback control via inertial manifolds for nonautonomous evolution equations. Communications on Pure and Applied Analysis, 2011, 10 (3) : 917-936. doi: 10.3934/cpaa.2011.10.917

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (119)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]