Attractors of differential inclusions and their approximation (Q2722241)

Let \((H,\langle \cdot,\cdot \rangle)\) be a separable Hilbert space, \(\varphi :H \mapsto(-\infty,\infty]\) be a proper, convex lower semi-continuous function with the domain \(D(\varphi)\), and \(\partial \varphi:D(\partial \varphi)\subset H \mapsto 2^H\) be the subdifferential of this function. The authors consider the following problemNEWLINE\[NEWLINE{dy(t)\over dt}\in -\partial \varphi(y(t))+F(y(t)), t\in [0,T], y(0)=y_0\in H,\tag{1}NEWLINE\]NEWLINE where \(F:H \mapsto 2^H\) is a multivalued mapping. A continuous function \(y(\cdot):[0,T] \mapsto H\) is called the strong solution of (1) if \(y(t)\) is absolutely continuous on each compact subset of \([0,T]\) and satisfies (1) almost everywhere. The set of such solutions gives rise to the mapping \(G:[0,\infty)\times X \mapsto 2^X:(t,y_0) \mapsto y(t),\) where \(X:=\text{cls} {D(\varphi)}\), which is called the multivalued semiflow. The authors establish conditions under which the semiflow \(G\) has a global, compact in X attractor \(\Xi \). The problem of approximation of the attractor \(\Xi \) by a sequence of attractors \(\Xi_n\) of semiflows \(G_n\) corresponding to equation (1) with \(F=F_n, \bigcap_{n=1}^\infty F_n(u)=F(u)\), is also studied.