These lecture notes are devoted to questions of the behaviour, when t → ∞, of trajectories Vt (ν), t ∈ ℔+ = [0, ∞) for semigroups {Vt, t ∈ ℔+, X} of nonlinear bounded continuous operators Vt in a locally non-compact metric space X and for solutions of abstract evolution equations. The latter contain many boundary value problems for PDE (partial differential equations) of a dissipative type.

In contrast to the traditional theory of the local stability of PDE (i.e. in the vicinity of a solution) we study the behaviour of all trajectories or solutions for the problems and give a description of the set of all limit states. We will not make assumptions either about the smallness of the parameters in the problem or on the closeness of the problem to a linear one, neither will we consider any other condition that ensures that all the solutions of the problem tend to some special solution. Our purpose is to develop a global theory of stability for problems of mathematical physics with dissipation. The principal ideas in this subject were formulated in paper [1] and I follow them here. The object of paper [1] concerns boundary value problems for Navier–Stokes equations. This object helped us to understand which properties of semigroup {Vt, t ∈ ℔+, X} imply the compactness of the set of all limit states (or, which is the same, the minimal global B-attractor), its invariance, the possibility of continuing the semigroup restricted on M. to the full group on M and a finiteness of dynamics {Vt, t ∈ ℔ = (−∞, + ∞), M}.