On the asymptotic behavior of the solutions to parabolic variational inequalities

TL;DR

This paper establishes a new constrained Lojasiewicz inequality for non-smooth obstacle problems, proving convergence of solutions and providing an abstract proof of the epiperimetric inequality for singular points.

Contribution

It introduces a novel constrained Lojasiewicz inequality applicable to non-analytic constraints and applies it to demonstrate solution convergence and to prove the epiperimetric inequality.

Findings

Solutions to obstacle problems converge to a unique stationary state.

A new inequality aids in analyzing non-analytic constraints.

The approach provides insights into singular points of obstacle problems.

Abstract

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.