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Smooth invariant foliations in infinite dimensional spaces. (English) Zbl 0749.58043

The authors consider the semilinear evolutionary equation \((du/dt)+Au=F(u,\lambda)\), where \(u\in Z\) and \(\lambda\in \Lambda\) is a parameter, with \(Z\) and \(\Lambda\) being Banach spaces. Under suitable hypotheses upon the linear operator \(A\) and the nonlinear term \(F\) one proves the existence and the uniqueness of a \(C^ k\) \(\gamma\)-unstable manifold \({\mathcal W}^ u_ \gamma\) and of an invariant foliation of the space \(Z^{\theta_ 1}\equiv\text{dom} (A+aI)^{\theta_ 1}\), for some constants \(a\) and \(\theta_ 1\), whose leaves are \(\gamma\)-stable and transverse to the invariant manifold \({\mathcal W}^ u_ \gamma\). These results imply the center-unstable manifold theorem and the inertial manifold theorem.
Reviewer: D.Motreanu (Iaşi)

MSC:

37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
58B10 Differentiability questions for infinite-dimensional manifolds
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