##
**Asymptotic behaviour and dynamics in infinite dimensions.**
*(English)*
Zbl 0653.35006

Nonlinear differential equations, Lect. 7th Congr., Granada/Spain 1984, Res. Notes Math. 132, 1-42 (1985).

[For the entire collection see Zbl 0638.00015.]

Let T(t) (t\(\geq 0)\) be a \(C^ r\)-semigroup on a Banach space. In this paper several conceptions, such as positive orbit \(\gamma^+(x)\), negative orbit \(\gamma^-(x)\), \(\omega\)-limit set \(\omega\) (x) of \(x\in X\), \(\alpha\)-limit set \(\alpha\) (x), global orbit, invariant set, and attractor, are defined. Then some conditions are imposed on T(t) to ensure that there is a compact attractor A and the stability properties of an attractor A are discussed. Considerable attention is also devoted to gradient systems, existence conditions for a compact attractor and the flow on the attractor. Finally, the dynamics of scalar parabolic equations are studied.

Let T(t) (t\(\geq 0)\) be a \(C^ r\)-semigroup on a Banach space. In this paper several conceptions, such as positive orbit \(\gamma^+(x)\), negative orbit \(\gamma^-(x)\), \(\omega\)-limit set \(\omega\) (x) of \(x\in X\), \(\alpha\)-limit set \(\alpha\) (x), global orbit, invariant set, and attractor, are defined. Then some conditions are imposed on T(t) to ensure that there is a compact attractor A and the stability properties of an attractor A are discussed. Considerable attention is also devoted to gradient systems, existence conditions for a compact attractor and the flow on the attractor. Finally, the dynamics of scalar parabolic equations are studied.

Reviewer: J.H.Tian

### MSC:

35B40 | Asymptotic behavior of solutions to PDEs |

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

35B35 | Stability in context of PDEs |

37C80 | Symmetries, equivariant dynamical systems (MSC2010) |

37D15 | Morse-Smale systems |

37C75 | Stability theory for smooth dynamical systems |