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

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  • 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.
    Mathematics Subject Classification: Primary: 34K19, 37D10; Secondary: 35R10.


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  • [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.


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


    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.


    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.


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


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


    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.


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


    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.


    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.


    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.


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


    J. L. 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, Integral Equations Operator Theory, 32 (1998), 332-353.doi: 10.1007/BF01203774.


    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.


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


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


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


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


    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.


    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.


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


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


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

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