Exponential global attractors for semigroups in metric spaces with applications to differential equations (Q2884092)

The authors of this very interesting paper consider asymptotically compact semigroups, that is a family \(\{S(t):t\geq 0\}\) of maps \(S(t):V\to V\) in some metric space \(V\) for which there exists a finite set of disjoint compact invariant sets attracting points of \(V\). The local unstable manifolds of these invariant sets are assumed to be pointwise exponentially attracting. More exactly it is assumed that: NEWLINE{\parindent=7mmNEWLINE\begin{itemize}\item[(i)]\(S(t)\) is an asymptotically compact and eventually continuous semigroup in \(V\); NEWLINE\item[(ii)]there is a finite set \(\mathcal{S}\) of disjoint compact invariant sets which attract points of \(V\); NEWLINE\item[(iii)]\(S(t)\) has eventually bounded orbits of bounded sets; NEWLINE\item[(iv)]\(S(t)\) satisfies a Lipschitz condition; NEWLINE\item[(v)]no element of \(\mathcal{S}\) is chain recurrent relative to \(\mathcal{S}\); NEWLINE\item[(vi)]each set \(Y^{\ast }\in \) \(\mathcal{S}\) has a pointwise exponentially attracting local unstable set. NEWLINENEWLINE\end{itemize}}NEWLINEUnder these assumptions \(S(t)\) has an invariant exponential global attractor. At the end of the paper the authors give some examples concerning ordinary differential equations and evolutionary problems by which they illustrate the theory.