American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation
  • DCDS-B Home
  • This Issue
  • Next Article
    Shape stability of optimal control problems in coefficients for coupled system of Hammerstein type
November  2015, 20(9): 2993-3011. doi: 10.3934/dcdsb.2015.20.2993

Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line

Nguyen Thieu Huy 1, and  Pham Van Bang 2,

1. 

School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi, Vietnam
2. 

Department of Basic Sciences, University of Economic and Technical Industries, 456-Minh Khai Str., Hai Ba Trung, Hanoi, Vietnam

Received  May 2014 Revised  July 2015 Published  September 2015

In this paper we investigate the existence of invariant stable and center-stable manifolds for solutions to partial neutral functional differential equations of the form $$\begin{cases}\frac{\partial}{\partial t}Fu_t = B(t)Fu_t + \Phi(t,u_t),\quad t\in (0,\infty),\cr u_0 = \phi\in \mathcal{C}: = C([-r, 0], X) \end{cases}$$ when the family of linear partial differential operators $(B(t))_{t\ge 0}$ generates the evolution family $(U(t,s))_{t\ge s\ge 0}$ (on Banach space $X$) having an exponential dichotomy or trichotomy on the half-line and the nonlinear delay operator $\Phi$ satisfies the $\varphi$-Lipschitz condition, i.e., $\| \Phi(t,\phi) -\Phi(t,\psi)\| \le \varphi(t)\|\phi -\psi\|_{\mathcal{C}}$ for $\phi, \psi\in \mathcal{C}$, where $\varphi(t)$ belongs to some admissible function space on the half-line.
Keywords: Exponential dichotomy, stable and center-stable manifolds, admissibility of function spaces., partial neutral functional differential equations.
Mathematics Subject Classification: Primary: 34K19, 37D10; Secondary: 35R1.
Citation: Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993
References:
[1]

B. Aulbach and N. V. Minh, Nonlinear semigroups and the existence and stability of semilinear nonautonomous evolution equations, Abstr. Appl. Anal., 1 (1996), 351-380. doi: 10.1155/S108533759600019X.  Google Scholar

[2]

P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations, Dyn. Rep., 2 (1989), 1-38.  Google Scholar

[3]

R. Benkhalti, K. Ezzinbi and S. Fatajou, Stable and unstable manifolds for nonlinear partial neutral functional differential equations, Diff. Integr. Eq., 23 (2010), 773-794.  Google Scholar

[4]

L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis, 34 (1998), 907-925. doi: 10.1016/S0362-546X(97)00569-5.  Google Scholar

[5]

J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences 35, Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[6]

C. Chicone, Ordinary Differential Equations with Applications, Springer-Verlag, 1999.  Google Scholar

[7]

I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, Journal of Dynamics and Differential Equations, 13 (2001), 355-380. doi: 10.1023/A:1016684108862.  Google Scholar

[8]

K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Text Math. 194, Springer-Verlag, New York, 2000.  Google Scholar

[9]

N. T. Huy, Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line, J. Math. Anal. Appl., 354 (2009), 372-386. doi: 10.1016/j.jmaa.2008.12.062.  Google Scholar

[10]

N. T. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235 (2006), 330-354. doi: 10.1016/j.jfa.2005.11.002.  Google Scholar

[11]

N. T. Huy and T. V. Duoc, Integral manifolds for partial functional differential equations in admissible spaces on a half-line, Journal of Mathematical Analysis and Applications, 411 (2014), 816-828. doi: 10.1016/j.jmaa.2013.10.027.  Google Scholar

[12]

R. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley Interscience, New York, 1976.  Google Scholar

[13]

J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Academic Press, New York, 1966.  Google Scholar

[14]

N. V. Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integral Equations Operator Theory, 32 (1998), 332-353. doi: 10.1007/BF01203774.  Google Scholar

[15]

N. V. Minh and J. Wu, Invariant manifolds of partial functional differential equations, J. Differential Equations, 198 (2004), 381-421. doi: 10.1016/j.jde.2003.10.006.  Google Scholar

[16]

J. D. Murray, Mathematical Biology I: An Introduction, Springer-Verlag, New York, 2002.  Google Scholar

[17]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2003.  Google Scholar

[18]

R. Nagel and G. Nickel, Well-posedness for non-autonomous abstract Cauchy problems, Progr. Nonlinear Differential Equations Appl., 50 (2002), 279-293.  Google Scholar

[19]

A. Pazy, Semigroup of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[20]

H. Petzeltová and O. J. Staffans, Spectral decomposition and invariant manifolds for some functional partial differential equations, J. Diff. Eq., 138 (1997), 301-327. doi: 10.1006/jdeq.1997.3277.  Google Scholar

[21]

F. Räbiger and R. Schnaubelt, The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions, Semigroup Forum, 52 (1996), 225-239. doi: 10.1007/BF02574098.  Google Scholar

[22]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, New York, Oxford, 1978.  Google Scholar

[23]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[24]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

show all references

References:
[1]

B. Aulbach and N. V. Minh, Nonlinear semigroups and the existence and stability of semilinear nonautonomous evolution equations, Abstr. Appl. Anal., 1 (1996), 351-380. doi: 10.1155/S108533759600019X.  Google Scholar

[2]

P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations, Dyn. Rep., 2 (1989), 1-38.  Google Scholar

[3]

R. Benkhalti, K. Ezzinbi and S. Fatajou, Stable and unstable manifolds for nonlinear partial neutral functional differential equations, Diff. Integr. Eq., 23 (2010), 773-794.  Google Scholar

[4]

L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis, 34 (1998), 907-925. doi: 10.1016/S0362-546X(97)00569-5.  Google Scholar

[5]

J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences 35, Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[6]

C. Chicone, Ordinary Differential Equations with Applications, Springer-Verlag, 1999.  Google Scholar

[7]

I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, Journal of Dynamics and Differential Equations, 13 (2001), 355-380. doi: 10.1023/A:1016684108862.  Google Scholar

[8]

K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Text Math. 194, Springer-Verlag, New York, 2000.  Google Scholar

[9]

N. T. Huy, Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line, J. Math. Anal. Appl., 354 (2009), 372-386. doi: 10.1016/j.jmaa.2008.12.062.  Google Scholar

[10]

N. T. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235 (2006), 330-354. doi: 10.1016/j.jfa.2005.11.002.  Google Scholar

[11]

N. T. Huy and T. V. Duoc, Integral manifolds for partial functional differential equations in admissible spaces on a half-line, Journal of Mathematical Analysis and Applications, 411 (2014), 816-828. doi: 10.1016/j.jmaa.2013.10.027.  Google Scholar

[12]

R. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley Interscience, New York, 1976.  Google Scholar

[13]

J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Academic Press, New York, 1966.  Google Scholar

[14]

N. V. Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integral Equations Operator Theory, 32 (1998), 332-353. doi: 10.1007/BF01203774.  Google Scholar

[15]

N. V. Minh and J. Wu, Invariant manifolds of partial functional differential equations, J. Differential Equations, 198 (2004), 381-421. doi: 10.1016/j.jde.2003.10.006.  Google Scholar

[16]

J. D. Murray, Mathematical Biology I: An Introduction, Springer-Verlag, New York, 2002.  Google Scholar

[17]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2003.  Google Scholar

[18]

R. Nagel and G. Nickel, Well-posedness for non-autonomous abstract Cauchy problems, Progr. Nonlinear Differential Equations Appl., 50 (2002), 279-293.  Google Scholar

[19]

A. Pazy, Semigroup of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[20]

H. Petzeltová and O. J. Staffans, Spectral decomposition and invariant manifolds for some functional partial differential equations, J. Diff. Eq., 138 (1997), 301-327. doi: 10.1006/jdeq.1997.3277.  Google Scholar

[21]

F. Räbiger and R. Schnaubelt, The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions, Semigroup Forum, 52 (1996), 225-239. doi: 10.1007/BF02574098.  Google Scholar

[22]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, New York, Oxford, 1978.  Google Scholar

[23]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[24]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

[1]

Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009
[2]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[3]

Éder Rítis Aragão Costa. An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 845-868. doi: 10.3934/cpaa.2019041
[4]

Alin Pogan, Kevin Zumbrun. Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks. Kinetic & Related Models, 2019, 12 (1) : 1-36. doi: 10.3934/krm.2019001
[5]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[6]

Luis Barreira, Claudia Valls. Characterization of stable manifolds for nonuniform exponential dichotomies. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1025-1046. doi: 10.3934/dcds.2008.21.1025
[7]

Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793
[8]

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

Luis Barreira, Claudia Valls. Stable manifolds with optimal regularity for difference equations. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1537-1555. doi: 10.3934/dcds.2012.32.1537
[10]

Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103
[11]

Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3579-3614. doi: 10.3934/dcds.2021008
[12]

Kai Liu, Zhi Li. Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3551-3573. doi: 10.3934/dcdsb.2016110
[13]

Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27
[14]

Mostafa Fazly, Yuan Li. Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4185-4206. doi: 10.3934/dcds.2021033
[15]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure & Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291
[16]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031
[17]

Jun Shen, Kening Lu, Bixiang Wang. Convergence and center manifolds for differential equations driven by colored noise. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4797-4840. doi: 10.3934/dcds.2019196
[18]

Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113
[19]

Vera Ignatenko. Homoclinic and stable periodic solutions for differential delay equations from physiology. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3637-3661. doi: 10.3934/dcds.2018157
[20]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences