Attractors, approximations and fixed sets of evolution systems (Q2894067)

Consider the evolution inclusion NEWLINE\[NEWLINE \dot{x} + A(x) \in F(x,u),\quad x(0)=a ,\quad u\in U,\quad t\in I, \tag{1} NEWLINE\]NEWLINE where the operator \(A:X\to X^{*},\;u\in U\) is a parameter, \(U\) is a compact metric space, \(I=[0,T]\). Here, \(X\subset H\subset X^{*}\), \(H\) is a separable Hilbert space, \(X\) is a separable and reflexive Banach space embedded continuously and densely into \(H\). Let \(Ex({\mathrm{\~A}},{\mathrm{\~B}}) =\sup_{a\in {\mathrm{\~A}}}\inf_{b\in {\mathrm{\~B}}}\{|a-b|\}\), where \({\mathrm{\~A}},{\mathrm{\~B}} \subset H\) are closed bounded sets. \(F\) has nonempty closed convex values, is bounded on bounded sets and satisfies a one sided Lipschitz condition with some negative constant.NEWLINENEWLINEThe reachable set of (1) is defined as NEWLINE\[NEWLINE\text{Reach}_{a}(t)=\{y\in H: \exists \text{ solution of (1) with } x(t) = y\}.NEWLINE\]NEWLINE The closed set \(V\subset H\) is said to be a strong attractor of (1) if and only ifNEWLINENEWLINE1) \( \mathrm{lim}_{t\to\infty} Ex(_{a}(t), V) =0\;\forall\, a\in H\);NEWLINENEWLINE2) \(\text{Reach}_{V}(t) = V\) for every \(t>0.\)NEWLINENEWLINEUnder some assumptions, the authors prove the existence of a unique strong forward attractor and a unique strong backward attractor.