Journal of Integral Equations and Applications

Stable and center-stable manifolds of admissible classes for partial functional differential equations

Trinh Viet Duoc and Nguyen Thieu Huy

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, we investigate the existence of stable and center-stable manifolds of admissible classes for mild solutions to partial functional differential equations of the form $\dot {u}(t)=A(t)u(t)+f(t,u_t)$, $t\ge 0$. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like $L_p$-spaces and many other function spaces occurring in interpolation theory such as the Lorentz spaces $L_{p,q}$. Results in this paper are the generalization and development for our results in \cite {HD1}. The existence of these manifolds obtained in the case that the family of operators $(A(t))_{t\ge 0}$ generate the evolution family $(U(t,s))_{t\ge s\ge 0}$ having an exponential dichotomy or trichotomy on the half-line and the nonlinear forcing term $f$ satisfies the $\varphi $-Lipschitz condition, i.e., $\| f(t,u_t) -f(t,v_t)\| \le \varphi (t)\|u_t -v_t\|_{\mathcal {C}}$, where $u_t,\ v_t \in \mathcal{C} :=C([-r, 0], X)$, and $\varphi (t)$ belongs to some admissible Banach function space and satisfies certain conditions.

Article information

Source
J. Integral Equations Applications, Volume 30, Number 4 (2018), 543-575.

Dates
First available in Project Euclid: 29 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1543482176

Digital Object Identifier
doi:10.1216/JIE-2018-30-4-543

Mathematical Reviews number (MathSciNet)
MR3881216

Zentralblatt MATH identifier
06989832

Subjects
Primary: 34K19: Invariant manifolds 34K25: Asymptotic theory 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx] 35B40: Asymptotic behavior of solutions

Keywords
Exponential dichotomy and trichotomy partial functional differential equations admissibility of function spaces stable and center-stable manifolds of admissible classes

Citation

Duoc, Trinh Viet; Huy, Nguyen Thieu. Stable and center-stable manifolds of admissible classes for partial functional differential equations. J. Integral Equations Applications 30 (2018), no. 4, 543--575. doi:10.1216/JIE-2018-30-4-543. https://projecteuclid.org/euclid.jiea/1543482176


Export citation

References

  • B. Aulbach, N.V. Minh, Nonlinear semigroups and the existence and stability of semilinear nonautonomous evolution equations, Abstr. Appl. Anal. 1 (1996), 351–380.
    Zentralblatt MATH: 0934.34051
    Digital Object Identifier: doi:10.1155/S108533759600019X
    Project Euclid: euclid.aaa/1049726080
  • P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations, Dynam. Rep. 2 (1989), 1–38.
    Zentralblatt MATH: 0674.58024
  • P. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc. 135 (1998), 645.
    Mathematical Reviews (MathSciNet): MR1445489
    Zentralblatt MATH: 1023.37013
    Digital Object Identifier: doi:10.1090/memo/0645
  • P. Bates, K. Lu and C. Zeng, Invariant foliations near normally hyperbolic invariant manifolds for semiflows, Trans. Amer. Math. Soc. 352 (2000), 4641–4676.
    Zentralblatt MATH: 0964.37018
    Digital Object Identifier: doi:10.1090/S0002-9947-00-02503-4
  • N. Bogoliubov and Yu. Mitropolsky, The method of integral manifolds in nonlinear mechanics, Contr. Diff. Eqs. 2 (1963), 123–196.
    Mathematical Reviews (MathSciNet): MR0149036
  • J. Carr, Applications of centre manifold theory, Appl. Math. Sci. 35 (1981).
    Zentralblatt MATH: 0464.58001
  • X.Y. Chen, J.K. Hale and B. Tan, Invariant foliations for $C^1$ semigroups in Banach spaces, J. Diff. Eqs. 139 (1997), 283–318.
  • C. Chicone and Y. Latushkin, Center manifolds for infinite dimensional nonautonomous differential equations, J. Diff Eqs. 141 (1997), 356–399.
    Zentralblatt MATH: 0992.34033
    Digital Object Identifier: doi:10.1006/jdeq.1997.3343
  • K.J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Grad. Text Math. 194, Springer-Verlag, Berlin, 2000.
    Zentralblatt MATH: 0952.47036
  • J.K. Hale, Flows on center manifolds for scalar functional differential equations, Proc. Roy. Soc. Edinburgh 101 (1985), 193–201.
    Mathematical Reviews (MathSciNet): MR824220
    Zentralblatt MATH: 0582.34058
    Digital Object Identifier: doi:10.1017/S030821050002076X
  • N. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lect. Notes Math. 183 (1977).
    Mathematical Reviews (MathSciNet): MR0501173
  • N.T. Huy, Invariant manifolds of admissible classes for semi-linear evolution equations, J. Diff. Eqs. 246 (2009), 1820–1844.
    Mathematical Reviews (MathSciNet): MR2494689
    Digital Object Identifier: doi:10.1016/j.jde.2008.10.010
  • ––––, Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line, J. Math. Anal. Appl. 354 (2009), 372–386.
  • ––––, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal. 235 (2006), 330–354.
  • N.T. Huy and T.V. Duoc, Integral manifolds for partial functional differential equations in admissible spaces on a half-line, J. Math. Anal. Appl. 411 (2014), 816–828.
    Mathematical Reviews (MathSciNet): MR3128434
    Zentralblatt MATH: 1332.34110
    Digital Object Identifier: doi:10.1016/j.jmaa.2013.10.027
  • ––––, Integral manifolds and their attraction property for evolution equations in admissible function spaces, Taiwanese J. Math. 16 (2012), 963–985.
    Mathematical Reviews (MathSciNet): MR2917249
    Digital Object Identifier: doi:10.11650/twjm/1500406669
    Project Euclid: euclid.twjm/1500406669
  • Y. Latushkin, J. Prüss and R. Schnaubelt, Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions, Discr. Contin. Dynam. Syst. 9 (2008), 595–633.
    Zentralblatt MATH: 1168.35021
    Digital Object Identifier: doi:10.3934/dcdsb.2008.9.595
  • J.J. Massera and J.J. Schäffer, Linear differential equations and function spaces, Academic Press, New York, 1966.
  • N.V. Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Int. Eqs. Oper. Th. 32 (1998), 332-353.
    Zentralblatt MATH: 0977.34056
    Digital Object Identifier: doi:10.1007/BF01203774
  • N.V. Minh and J. Wu, Invariant manifolds of partial functional differential equations, J. Diff. Eqs. 198 (2004), 381–421.
  • J.D. Murray, Mathematical biology I: An introduction, Springer-Verlag Berlin, 2002.
    Zentralblatt MATH: 1006.92001
  • ––––, Mathematical biology II: Spatial models and biomedical applications, Springer-Verlag, Berlin, 2003.
  • F. Räbiger and R. Schnaubelt, The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions, Semigr. Forum 8 (1996), 225–239.
    Mathematical Reviews (MathSciNet): MR1371805
    Digital Object Identifier: doi:10.1007/BF02574098
  • R. Schnaubelt, Asymptotically autonomous parabolic evolution equations, J. Evol. Eqs. 1 (2001), 19–37.
    Zentralblatt MATH: 1098.34551
    Digital Object Identifier: doi:10.1007/PL00001363
  • J. Wu, Theory and applications of partial functional differential equations, Springer Verlag, Berlin, 1996.
    Zentralblatt MATH: 0870.35116
  • A. Yagi, Abstract parabolic evolution equations and their applications, Springer Verlag, New York, 2009.
    Zentralblatt MATH: 1190.35004

Rocky Mountain Mathematics Consortium

Journal of Integral Equations and Applications

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more