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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

 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.
Citation: Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete and 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. [2] P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations, Dyn. Rep., 2 (1989), 1-38. [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. [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. [5] J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences 35, Springer-Verlag, New York-Berlin, 1981. [6] C. Chicone, Ordinary Differential Equations with Applications, Springer-Verlag, 1999. [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. [8] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Text Math. 194, Springer-Verlag, New York, 2000. [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. [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. [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. [12] R. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley Interscience, New York, 1976. [13] J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Academic Press, New York, 1966. [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. [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. [16] J. D. Murray, Mathematical Biology I: An Introduction, Springer-Verlag, New York, 2002. [17] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2003. [18] R. Nagel and G. Nickel, Well-posedness for non-autonomous abstract Cauchy problems, Progr. Nonlinear Differential Equations Appl., 50 (2002), 279-293. [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. [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. [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. [22] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, New York, Oxford, 1978. [23] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [24] 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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##### 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. [2] P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations, Dyn. Rep., 2 (1989), 1-38. [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. [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. [5] J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences 35, Springer-Verlag, New York-Berlin, 1981. [6] C. Chicone, Ordinary Differential Equations with Applications, Springer-Verlag, 1999. [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. [8] K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Text Math. 194, Springer-Verlag, New York, 2000. [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. [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. [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. [12] R. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley Interscience, New York, 1976. [13] J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Academic Press, New York, 1966. [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. [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. [16] J. D. Murray, Mathematical Biology I: An Introduction, Springer-Verlag, New York, 2002. [17] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer-Verlag, New York, 2003. [18] R. Nagel and G. Nickel, Well-posedness for non-autonomous abstract Cauchy problems, Progr. Nonlinear Differential Equations Appl., 50 (2002), 279-293. [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. [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. [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. [22] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, New York, Oxford, 1978. [23] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [24] 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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