  We develop the long-time analysis for gradient flow equations in metric
spaces. In particular, we consider two notions of solutions for metric gradient
flows, namely energy and generalized solutions. While the former concept
coincides with the notion of curves of maximal slope, we introduce the latter
to include limits of time-incremental approximations constructed via the
Minimizing Movements approach.
  For both notions of solutions we prove the existence of the global attractor.
Since the evolutionary problems we consider may lack uniqueness, we rely on the
theory of generalized semiflows introduced by J.M. Ball. The notions of
generalized and energy solutions are quite flexible and can be used to address
gradient flows in a variety of contexts, ranging from Banach spaces to
Wasserstein spaces of probability measures.
  We present applications of our abstract results by proving the existence of
the global attractor for the energy solutions both of abstract doubly nonlinear
evolution equations in reflexive Banach spaces, and of a class of evolution
equations in Wasserstein spaces, as well as for the generalized solutions of
some phase-change evolutions driven by mean curvature.
