a global attractor
Run a dissipative evolution equation for a very long time, starting from many different initial states. Energy bleeds away, transients die out, and whatever survives — the eventual repertoire of long-time behaviours — collapses onto a single, comparatively small set that every trajectory is drawn toward. That terminal set is the global attractor. It is the geometric object that captures 'where the dynamics ultimately lives', the long-time skeleton of an infinite-dimensional system, and it is the central object of the theory of attractors for dissipative PDEs.
Precisely, view the solution operators as a semigroup S(t) on a Banach space X (S(t) u_0 = u(t) is the state at time t starting from u_0). The global attractor is a set Ac in X that is compact, invariant (S(t) Ac = Ac for all t — it maps onto itself, so trajectories on it stay on it forever, forward and backward), and attracting: for every bounded set of initial data B, the distance from S(t) B to Ac goes to 0 as t goes to infinity. Compact and attracting together force Ac to be the smallest such set and to contain all the 'permanent' dynamics — all steady states, all periodic orbits, and the connecting trajectories between them. The standard existence recipe is: prove the semigroup is dissipative (there is a bounded absorbing set every orbit eventually enters) and asymptotically compact (the smoothing of a parabolic equation makes the dynamics effectively finite-dimensional in the limit); then the attractor exists and equals the omega-limit set of the absorbing set.
Why it matters: a global attractor compresses the long-time behaviour of an infinite-dimensional PDE into one compact set, often of finite fractional dimension despite the phase space being infinite-dimensional — this is the rigorous sense in which a turbulent or pattern-forming system has 'only finitely many degrees of freedom in the long run'. For the simplest gradient-like parabolic equations the attractor is just the set of steady states plus the heteroclinic orbits joining them, so studying long-time behaviour reduces to studying the elliptic steady-state problem and its stability. When there is a spectral gap and a unique stable equilibrium, the attractor degenerates to that single point.
For u_t = Laplacian u with Dirichlet data on a bounded domain, every solution decays to zero (spectral gap lambda_1 > 0), so the global attractor is the single point {0}. For a bistable reaction-diffusion equation u_t = Laplacian u + u - u^3, the attractor is larger: it contains the constant steady states +1 and -1, the unstable state 0, and the front solutions that connect them.
The attractor collapses long-time dynamics onto a small invariant set — sometimes a single equilibrium.
A global attractor describes the long-time limit set, not how fast you reach it, and trajectories can wander on a complicated attractor forever without settling to a point — 'attracting' does not mean 'static'. Existence requires dissipativity; conservative equations (like the undamped wave equation, which preserves energy) have no global attractor at all.