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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 $X\sb 0$, $T: [0,+\infty [\to C(X,X)$ be a $C\sp 0$-semigroup such that $T(t)(X\sb 0)\subset X\sb 0$, $T(t)(\partial X\sb 0)\subset \partial X\sb 0$. A subset U of $X\sb 0$ is said strongly bounded if it is bounded and there is $\eta >0$ such that $d(x,\partial X\sb 0)\ge \eta$ for $x\in U$; a subset A of X [of $X\sb 0]$ is said a global attractor [relative to strongly bounded sets] if it is compact, $T(t)A=a$ for $t\ge 0$ and $\lim\sb{t\to +\infty}\delta (T(t)B,A)=0$ for any [strongly] bounded subset B of X $[X\sb 0]$, 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 $t\sb 0(x,B)$ such that $T(t)x\in B$ for $t\ge t\sb 0(x,B)$, asymptotically smooth if for any bounded subset B of X, for which T(t)B$\subset B$ for $t\ge 0$, there exists a compact set K such that $\lim\sb{t\to +\infty}\delta (T(t)B,K)=0,$ uniformly persistent if there is $\eta >0$ such that $\liminf\sb{t\to +\infty}d(T(t)x,\partial X\sb 0)\ge \eta$ for $x\in X\sb 0.$ Then the authors prove: If $T$ is point dissipative and uniformly persistent and (i) there is a $t\sb 0\ge 0$ such that $T(t)$ is compact for $t>t\sb 0$ or (ii) T is asymptotically smooth, $\cup \{T(t)U:$ $t\ge 0\}$ is bounded if U is bounded in $X$, $\{T(t)V, t\ge 0\}$ is strongly bounded if V is strongly bounded, then there are global attractors A in X and $A\sb{\partial}$ in $\partial X\sb 0$ and a global attractor $A\sb 0$ in $X\sb 0$ relative to strongly bounded sets. Furthermore $A=A\sb 0\cup W\sp u(A\sb{\partial})$, where $$W\sp u(A\sb{\partial})=\{x\in A:\quad \cap \{cl\{\phi (\theta)\in X:\quad \theta \le -\tau \text{ and } \phi: ]-\infty,0]\to A,\quad \phi (0)=x,\quad T(t)\phi (s)=\phi (t+s)$$ $$\text{for every }s,t:\quad s\le -t\le 0\},\quad t\ge 0\}\subset A\sb{\partial}\}.$$ Let T be point dissipative; there exist pairwise disjoint, compact sets $M\sb 1,...,M\sb k$ such that $M\sb i$ is invariant (i.e. $T(t)M\sb i=M\sb i$ for $t\ge 0)$ set and is the maximal invariant set of a neighbourhood of itself for $T(t)\vert\sb{\partial X\sb 0}$ and for $T(t)$ for $t\ge 0$ and $i=1,...,k$, $\cup\sp{k}\sb{i=1}M\sb k\supset \cup \{\omega (X)$, $x\in A\sb{\partial}\}$, where $\omega (x)=\cap \overline{\{\cup \{T(t)x:t\ge \tau \}}:\tau\ge 0\}$, don’t exist $j\sb 1,...,j\sb h\in \{1,...,k\}$ such that $j\sb 1=j\sb h$, there exists $X\not\in M\sb{j\sb i}\cup M\sb{j\sb{i+1}}$ such that $x\in W\sp u(M\sb{j\sb i})\cap W\sp s(M\sb{j\sb{i+1}})$ for $i=1,...,h-1$, where $W\sp s(A)=\{x\in X:$ $\omega(x)\ne \emptyset$, $\omega(x)\subset A\}$; (i) there is a $t\sb 0\ge 0$ such that T(t) is compact for $t>t\sb 0$ or (ii) T is asymptotically smooth, $\cup \{T(t)U:$ $t\ge 0\}$ is bounded if U is bounded in X. Then T is uniformly persitent if and only if $W\sp s(M\sb i)\cap X\sb 0=\emptyset$ for each $i\in \{1,...,k\}$.
Reviewer: G.Bottaro

##### MSC:
 37-99 Dynamic systems and ergodic theory (MSC2000) 34G20 Nonlinear ODE in abstract spaces
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