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Acta Mathematica Vietnamica

, Volume 42, Issue 1, pp 187–207 | Cite as

Unstable Manifolds for Partial Neutral Differential Equations and Admissibility of Function Spaces

  • Thieu Huy NguyenEmail author
  • Van Bang Pham
Article
  • 134 Downloads

Abstract

We prove the existence and attraction property of an unstable manifold for solutions to the partial neutral functional differential equation of the form

\(\left\{\begin{array}{ll} \frac{\partial}{\partial t}Fu_{t}= B(t)Fu_{t} +\varPhi(t,u_{t}),\quad t\ge s;~ t,s\in \mathbb{R},\\ u_{s}=\phi\in \mathcal{C}:=C([-r, 0], X) \end{array}\right.\)

under the conditions that the family of linear operators \((B(t))_{t\in \mathbb {R}}\) defined on a Banach space X generates the evolution family (U(t, s)) ts having an exponential dichotomy on the whole line \(\mathbb {R}\), the difference operator \(F:\mathcal {C}\to X\) is bounded and linear, and the nonlinear delay operator Φ satisfies the ϕ-Lipschitz condition, i.e., \(\| \Phi (t,\phi ) -\Phi (t,\psi )\| \le \phi (t)\|\phi -\psi \|_{\mathcal {C}}\) for \(\phi ,~ \psi \in \mathcal {C}\), where ϕ(⋅) belongs to an admissible function space defined on \(\mathbb {R}\). Our main method is based on Lyapunov-Perron’s equations combined with the admissibility of function spaces and the technique of choosing F-induced trajectories.

Keywords

Exponential dichotomy Partial neutral functional differential equations Unstable manifolds Attractiveness Admissibility of function spaces 

Mathematics Subject Classification (2010)

34K19 35R10 

Notes

Acknowledgments

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant No. 101.02-2014.02.

References

  1. 1.
    Aulbach, B., Minh, N.V.: Nonlinear semigroups and the existence and stability of semilinear nonautonomous evolution equations. Abstr. Appl. Anal. 1, 351–380 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benkhalti, R., Ezzinbi, K., Fatajou, S.: Stable and unstable manifolds for nonlinear partial neutral functional differential equations. Differ. Integr. Equa. 23, 601–799 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Huy, N.T.: Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line. J. Funct. Anal. 235, 330–354 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Huy, N.T.: Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line. J. Math. Anal. Appl. 354, 372–386 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Huy, N.T., Bang, P.V.: Dichotomy and positivity of neutral equations with nonautonomous past. Appl. Anal. Discrete Math. 8, 224–242 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Huy, N.T., Bang, P.V.: Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete and Continuous Dynamical Systems Series B 20, 2993–3011 (2015)Google Scholar
  7. 7.
    Massera, J.J., Schäffer, J.J.: Linear Differential Equations and Function Spaces. Academic Press, New York (1966)zbMATHGoogle Scholar
  8. 8.
    Minh, N.V., Räbiger, F.R., Schnaubelt, R.: Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line. Integr. Equ. Oper. Theory 32, 332–353 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Minh, N.V., Wu, J.: Invariant manifolds of partial functional differential equations. J. Differ. Equa. 198, 381–421 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Murray, J.D.: Mathematical Biology I: An Introduction. Springer, Berlin (2002)zbMATHGoogle Scholar
  11. 11.
    Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, Berlin (2003)zbMATHGoogle Scholar
  12. 12.
    Nagel, R., Nickel, G.: Well-posedness for non-autonomous abstract Cauchy problems. Progr. Nonlinear Differential Equations Appl. 50, 279–293 (2002)zbMATHGoogle Scholar
  13. 13.
    Pazy, A.: Semigroup of Linear Operators and Application to Partial Differential Equations. Springer, Berlin (1983)Google Scholar
  14. 14.
    Petzeltová, H., Staffans, O.J.: Spectral decomposition and invariant manifolds for some functional partial differential equations. J. Differ. Equa. 138, 301–327 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Räbiger, F., Schnaubelt, R.: The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions. Semigroup Forum 48, 225–239 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer (1996)Google Scholar
  17. 17.
    Yagi, A.: Abstract Parabolic Evolution Equations and their Applications. Springer (2009)Google Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017

Authors and Affiliations

  1. 1.School of Applied Mathematics and InformaticsHanoi University of Science and TechnologyHanoiVietnam
  2. 2.Department of Basic SciencesUniversity of Economic and Technical IndustriesHanoiVietnam

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