We prove the existence of integral (stable, unstable, and center) manifolds for the solutions to a semilinear integral equation
in the case where the evolution family (U(t, s)) t≥s has an exponential trichotomy on a half line or on the whole line, and the nonlinear forcing term f satisfies the φ-Lipschitz conditions, i.e.,
where φ(t) belongs to some classes of admissible function spaces. Our main method is based on the Lyapunov–Perron methods, rescaling procedures, and the techniques of using the admissibility of function spaces.
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References
D. Anosov, “Geodesic flows on closed Riemann manifolds with negative curvature,” Proc. Steklov Inst. Math., 90 (1967).
B. Aulbach and N. V. Minh, “Nonlinear semigroups and the existence and stability of semilinear nonautonomous evolution equations,” Abstract Appl. Anal., 1, 351–380 (1996).
P. Bates and C. Jones, “Invariant manifolds for semilinear partial differential equations,” Dynam. Rep., 2, 1–38 (1989).
N. Bogoliubov and Yu. Mitropolsky, “The method of integral manifolds in nonlinear mechanics,” Contrib. Different. Equat., 2, 123–196 (1963).
N. Bogoliubov and Yu. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York (1961).
A. P. Calderon, “Spaces between L 1 and L ∞ and the theorem of Marcinkiewicz,” Stud. Math., 26, 273–299 (1996).
J. Carr, Applications of Centre Manifold Theory, Springer, New York–Berlin (1981).
J. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces, American Mathematical Society, Providence, RI (1974).
K. J. Engel and R. Nagel, “One-parameter semigroups for linear evolution equations,” Grad. Text Math., 194 (2000).
J. Hadamard, “Sur l’itération et les solutions asymptotiques des equations diff´erentielles,” Bull. Soc. Math. France, 29, 224–228 (1991).
J. K. Hale, L. T. Magalhães, and W. M. Oliva, Dynamics in Infinite Dimensions, Springer, New York (2002).
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin (1981).
M. Hirsch, C. Pugh, and M. Shub, Invariant Manifolds, Springer, Berlin (1977).
Nguyen Thieu Huy, “Exponential dichotomy of evolution equations and admissibility of function spaces on a half line,” J. Funct. Anal., 235, 330–354 (2006).
Nguyen Thieu Huy, “Stable manifolds for semilinear evolution equations and admissibility of function spaces on a half line,” J. Math. Anal. Appl., 354, 372–386 (2009).
Nguyen Thieu Huy, “Invariant manifolds of admissible classes for semilinear evolution equations,” J. Different. Equat., 246, 1820–1844 (2009).
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II. Function Spaces, Springer, Berlin (1979).
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel (1995).
R. H. Martin, Jr., Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York (1976).
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,” Integr. Equat. Oper. Theory, 32, 332–353 (1998).
N. V. Minh and J. Wu, “Invariant manifolds of partial functional differential equations,” J. Different. Equat., 198, 381–421 (2004).
J. D. Murray, Mathematical Biology I: An Introduction, Springer, Berlin (2002).
J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, Berlin (2003).
R. Nagel and G. Nickel, “Well-posedness of nonautonomous abstract Cauchy problems,” Prog. Nonlin. Different. Equat. Appl., 50, 279–293 (2002).
G. Nickel, On Evolution Semigroups and Well-Posedness of Nonautonomous Cauchy Problems, PhD Thesis, Tübingen (1996).
Z. Nitecki, An Introduction to the Orbit Structure of Diffeomorphisms, MIT Press, Cambridge, MA (1971).
A. Pazy, Semigroup of Linear Operators and Application to Partial Differential Equations, Springer, Berlin (1983).
O. Perron, “Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen,” Math. Z., 29, No. 1, 129–160 (1929).
O. Perron, “Die Stabilitätsfrage bei Differentialgleichungen,” Math. Z., 32, 703–728 (1930).
F. Räbiger and R. Schaubelt, “The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions,” Semigroup Forum, 48, 225–239 (1996).
R. Schnaubelt, Exponential Bounds and Hyperbolicity of Evolution Families, PhD Thesis, Tübingen (1996).
R. Schnaubelt, Exponential Dichotomy of Nonautonomous Evolution Equations, Habilitationsschrift, Tübingen (1999).
R. Schnaubelt, “Asymptotically autonomous parabolic evolution equations,” J. Evol. Equat., 1, 19–37 (2001).
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York (2002).
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam (1978).
A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer, Berlin (2009).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 6, pp. 772–796, June, 2012.
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Huy, N.T., Ha, V.T.N. & Phi, H. Integral manifolds for semilinear evolution equations and admissibility of function spaces. Ukr Math J 64, 881–911 (2012). https://doi.org/10.1007/s11253-012-0686-1
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DOI: https://doi.org/10.1007/s11253-012-0686-1