  This paper is dedicated to providing new tools and methods for studying the
trend to equilibrium of gradient flows in metric spaces in the entropy and
metric sense, to establish decay rates, finite time of extinction, and to
characterize Lyapunov stable equilibrium points. In addition we outline that
the celebrated Entropy-Entropy production inequality used in kinetic theory is
nothing less than a global Kurdyka-Lojasiewicz-Simon inequality. This links two
different areas, namely, algebraic geometry with kinetic theory. As an
application of the tools developed in this paper, we obtain the following
results:
  - New upper bounds on the extinction time of gradient flows associated with
the total variational flow.
  - If the metric space is the p-Wasserstein space, then new HWI-, Talagrand-,
and logarithmic Sobolev inequalities are obtained for functionals associated
with nonlinear diffusion problems modeling drift, potential and interaction
phenomena.
  - It is shown that these inequalities are equivalent to the
Kurdyka-Lojasiewicz-Simon inequality and hence, they imply trend to equilibrium
of the gradient flows with decay rates or arrival in finite time.
