On the asymptotic behavior of the solutions to parabolic variational inequalities

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 Łojasiewicz inequality. The difficulty lies in the fact that, since the constraint is non-analytic, the pioneering method of L. Simon ([22]) 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.