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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).

37C80 Symmetries, equivariant dynamical systems (MSC2010)
35K55 Nonlinear parabolic equations
35L70 Second-order nonlinear hyperbolic equations
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