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Persistence in infinite-dimensional systems. (English) Zbl 0692.34053

Let \((X,d)\) be a complete metric space which is the closure of an open set \(X_ 0\), \(T: [0,+\infty [\to C(X,X)\) be a \(C^ 0\)-semigroup such that \(T(t)(X_ 0)\subset X_ 0\), \(T(t)(\partial X_ 0)\subset \partial X_ 0\). A subset U of \(X_ 0\) is said strongly bounded if it is bounded and there is \(\eta >0\) such that \(d(x,\partial X_ 0)\geq \eta\) for \(x\in U\); a subset A of X [of \(X_ 0]\) is said a global attractor [relative to strongly bounded sets] if it is compact, \(T(t)A=a\) for \(t\geq 0\) and \(\lim_{t\to +\infty}\delta (T(t)B,A)=0\) for any [strongly] bounded subset B of X \([X_ 0]\), where \(\delta (B,A)=\sup \{\inf \{d(y,x):\;x\in A\}:\;y\in B\}\), T is said point dissipative if there is a bounded non-empty set B in X such that for any \(x\in X\) there is \(t_ 0(x,B)\) such that \(T(t)x\in B\) for \(t\geq t_ 0(x,B)\), asymptotically smooth if for any bounded subset B of X, for which T(t)B\(\subset B\) for \(t\geq 0\), there exists a compact set K such that \(\lim_{t\to +\infty}\delta (T(t)B,K)=0,\) uniformly persistent if there is \(\eta >0\) such that \(\liminf_{t\to +\infty}d(T(t)x,\partial X_ 0)\geq \eta\) for \(x\in X_ 0.\)
Then the authors prove: If \(T\) is point dissipative and uniformly persistent and (i) there is a \(t_ 0\geq 0\) such that \(T(t)\) is compact for \(t>t_ 0\) or (ii) T is asymptotically smooth, \(\cup \{T(t)U:\) \(t\geq 0\}\) is bounded if U is bounded in \(X\), \(\{T(t)V, t\geq 0\}\) is strongly bounded if V is strongly bounded, then there are global attractors A in X and \(A_{\partial}\) in \(\partial X_ 0\) and a global attractor \(A_ 0\) in \(X_ 0\) relative to strongly bounded sets. Furthermore \(A=A_ 0\cup W^ u(A_{\partial})\), where \[ W^ u(A_{\partial})=\{x\in A:\quad \cap \{cl\{\phi (\theta)\in X:\quad \theta \leq -\tau \text{ and } \phi: ]-\infty,0]\to A,\quad \phi (0)=x,\quad T(t)\phi (s)=\phi (t+s) \]
\[ \text{for every }s,t:\quad s\leq -t\leq 0\},\quad t\geq 0\}\subset A_{\partial}\}. \] Let T be point dissipative; there exist pairwise disjoint, compact sets \(M_ 1,...,M_ k\) such that \(M_ i\) is invariant (i.e. \(T(t)M_ i=M_ i\) for \(t\geq 0)\) set and is the maximal invariant set of a neighbourhood of itself for \(T(t)|_{\partial X_ 0}\) and for \(T(t)\) for \(t\geq 0\) and \(i=1,...,k\), \(\cup^{k}_{i=1}M_ k\supset \cup \{\omega (X)\), \(x\in A_{\partial}\}\), where \(\omega (x)=\cap \overline{\{\cup \{T(t)x:t\geq \tau \}}:\tau\geq 0\}\), don’t exist \(j_ 1,...,j_ h\in \{1,...,k\}\) such that \(j_ 1=j_ h\), there exists \(X\not\in M_{j_ i}\cup M_{j_{i+1}}\) such that \(x\in W^ u(M_{j_ i})\cap W^ s(M_{j_{i+1}})\) for \(i=1,...,h-1\), where \(W^ s(A)=\{x\in X:\) \(\omega(x)\neq \emptyset\), \(\omega(x)\subset A\}\); (i) there is a \(t_ 0\geq 0\) such that T(t) is compact for \(t>t_ 0\) or (ii) T is asymptotically smooth, \(\cup \{T(t)U:\) \(t\geq 0\}\) is bounded if U is bounded in X. Then T is uniformly persitent if and only if \(W^ s(M_ i)\cap X_ 0=\emptyset\) for each \(i\in \{1,...,k\}\).
Reviewer: G.Bottaro

MSC:

37-XX Dynamical systems and ergodic theory
34G20 Nonlinear differential equations in abstract spaces
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