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Invariant manifolds for flows in Banach spaces. (English) Zbl 0691.58034

We present a theory of smooth invariant manifolds based on the classical method of Lyapunov-Perron for continuous semiflows in Banach spaces. Basic hypotheses for these semiflows will be satisfied by semilinear parabolic equations on bounded or unbounded domains or hyperbolic equations. Examples of these continuous semigroups from evolution equations may be found in P. Bates and C. Jones [The center manifold theorem with applications (preprint)]. The two basic theorems are stated for nonlinear integral equations. One is on the existence of smooth invariant manifolds (Theorem 4.4) and the other is on exponential attractivity of invariant manifolds (Theorem 5.1).

MSC:

37C80 Symmetries, equivariant dynamical systems (MSC2010)
35K55 Nonlinear parabolic equations
35L70 Second-order nonlinear hyperbolic equations
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[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101
[2] Babin, A. V.; Vishik, M. I., Unstable invariant sets of semigroups of nonlinear operators and their perturbations, Russian Math. Surveys, 41, 1-41 (1986) · Zbl 0624.47065
[4] Carr, J., Application of Center Manifold Theory, (Applied Mathematical Sciences, Vol. 35 (1981), Springer-Verlag: Springer-Verlag New York)
[5] Chow, S.-N; Hale, J. K., Methods of Bifurcation Theory (1982), Springer-Verlag: Springer-Verlag New York
[9] Constantin, P.; Foias, C.; Temam, R., Attractors representing turbulent flows, Mem. Amer. Math. Soc., 314 (1985) · Zbl 0567.35070
[10] Conway, E.; Hoff, D.; Smoller, J., Large time behavior of solutions of non-linear reaction-diffusion equations, SIAM J. Appl. Math., 35, 1-16 (1978) · Zbl 0383.35035
[12] Foias, C.; Nicolaenko, B.; Sell, G. R.; Temam, R., Variétés inertielles pour l’équation de Kuramoto-Sivashinsky, C. R. Acad. Sci. Paris, Sér. I Math., 301, 285-288 (1985) · Zbl 0591.35063
[15] van Gils, S. A.; Vanderbauwhede, A., Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal., 72, 209-224 (1987) · Zbl 0621.47050
[16] Hale, J. K., Theory of Functional Differential Equations (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0425.34048
[18] Hale, J. K.; Lin, X. B., Symbolic dynamics and nonlinear flows, Ann. Mat. Pura Appl. (4), 144, 229-260 (1986)
[19] Hale, J. K.; Magalhaes, L. T.; Oliva, W. M., An Introduction to Infinite Dimensional Dynamical Systems-Geometric Theory, (Appl. Math. Sciences, Vol. 47 (1984), Springer-Verlag: Springer-Verlag New York) · Zbl 0533.58001
[21] Henry, D., Geometric Theory of Parabolic Equation, (Lecture Notes in Math., Vol. 840 (1981), Springer-Verlag: Springer-Verlag New York)
[22] Hirsch, M.; Pugh, C., Stable manifolds and hyperbolic sets, (Proc. Sympos. Pure Math., 14 (1970)), 133-1163 · Zbl 0215.53001
[23] Mallet-Paret, J., Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, J. Differential Equations, 22, 331-348 (1976) · Zbl 0354.34072
[25] Mane, R., On the Dimension of the Compact Invariant Sets of Certain Nonlinear Maps, (Lecture Notes in Math., Vol. 898 (1981), Springer-Verlag: Springer-Verlag New York), 230-242
[27] Nicolaenko, B.; Scheurer, B.; Temam, R., Some global dynamical properties of the Kuramoto Sivashinsky equations: Nonlinear stability and attractors, Phys. D, 16, 155-183 (1985) · Zbl 0592.35013
[29] Mora, X.; Sola-Morales, J., Existence and nonexistence of finite dimensional globally attracting invariant manifolds in semilinear damped wave equations, (Chow, S. N.; Hale, J. K., Dynamics on Infinite Dimensional Systems (1987), Springer-Verlag: Springer-Verlag New York) · Zbl 0642.35062
[31] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, (Applied Mathematical Sciences, Vol. 44 (1983), Springer-Verlag: Springer-Verlag New York) · Zbl 0516.47023
[32] Wells, J. C., Invariant manifolds of nonlinear operators, Pacific J. Math., 62, 285-293 (1976) · Zbl 0343.58010
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