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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
Nguyen Thieu Huy 1, and Pham Van Bang 2,
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi, Vietnam
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
B. Aulbach and N. V. Minh, Nonlinear semigroups and the existence and stability of semilinear nonautonomous evolution equations,, Abstr. Appl. Anal., 1 (1996), 351. doi: 10.1155/S108533759600019X. Google Scholar
P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations,, Dyn. Rep., 2 (1989), 1. Google Scholar
R. Benkhalti, K. Ezzinbi and S. Fatajou, Stable and unstable manifolds for nonlinear partial neutral functional differential equations,, Diff. Integr. Eq., 23 (2010), 773. Google Scholar
L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations,, Nonlinear Analysis, 34 (1998), 907. doi: 10.1016/S0362-546X(97)00569-5. Google Scholar
J. Carr, Applications of Centre Manifold Theory,, Applied Mathematical Sciences 35, (1981). Google Scholar
C. Chicone, Ordinary Differential Equations with Applications,, Springer-Verlag, (1999). Google Scholar
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. doi: 10.1023/A:1016684108862. Google Scholar
K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations,, Graduate Text Math. 194, (2000). Google Scholar
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. doi: 10.1016/j.jmaa.2008.12.062. Google Scholar
N. T. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line,, J. Funct. Anal., 235 (2006), 330. doi: 10.1016/j.jfa.2005.11.002. Google Scholar
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. doi: 10.1016/j.jmaa.2013.10.027. Google Scholar
R. Martin, Nonlinear Operators and Differential Equations in Banach Spaces,, Wiley Interscience, (1976). Google Scholar
J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces,, Academic Press, (1966). Google Scholar
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. doi: 10.1007/BF01203774. Google Scholar
N. V. Minh and J. Wu, Invariant manifolds of partial functional differential equations,, J. Differential Equations, 198 (2004), 381. doi: 10.1016/j.jde.2003.10.006. Google Scholar
J. D. Murray, Mathematical Biology I: An Introduction,, Springer-Verlag, (2002). Google Scholar
J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications,, Springer-Verlag, (2003). Google Scholar
R. Nagel and G. Nickel, Well-posedness for non-autonomous abstract Cauchy problems,, Progr. Nonlinear Differential Equations Appl., 50 (2002), 279. Google Scholar
A. Pazy, Semigroup of Linear Operators and Application to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar
H. Petzeltová and O. J. Staffans, Spectral decomposition and invariant manifolds for some functional partial differential equations,, J. Diff. Eq., 138 (1997), 301. doi: 10.1006/jdeq.1997.3277. Google Scholar
F. Räbiger and R. Schnaubelt, The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions,, Semigroup Forum, 52 (1996), 225. doi: 10.1007/BF02574098. Google Scholar
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978). Google Scholar
J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer Verlag, (1996). doi: 10.1007/978-1-4612-4050-1. Google Scholar
A. Yagi, Abstract Parabolic Evolution Equations and their Applications,, Springer-Verlag, (2010). doi: 10.1007/978-3-642-04631-5. Google Scholar
Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009
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
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