  We consider various versions of the obstacle and thin-obstacle problems, we
interpret them as variational inequalities, with non-smooth constraint, and
prove that they satisfy a new constrained Lojasiewicz inequality. The
difficulty lies in the fact that, since the constraint is non-analytic, the
pioneering method of L. Simon (Ann. of Math. 118(3), 1983) does not apply and
we have to exploit a better understanding on the constraint itself. We then
apply this inequality to two associated problems. First we combine it with an
abstract result on parabolic variational inequalities, to prove the convergence
at infinity of the strong global solutions to the parabolic obstacle and
thin-obstacle problems to a unique stationary solution with a rate. Secondly,
we give an abstract proof, based on a parabolic approach, of the epiperimetric
inequality, which we then apply to the singular points of the obstacle and
thin-obstacle problems.
