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

Hale, J. K.

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.

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

Keywords:

asymptotic behaviour; infinite dimensions; dynamical systems; attractor; \(C^ r\)-semigroup; Banach space; compact attractor; stability; gradient systems; existence; flow; dynamics

Citations:

Zbl 0638.00015