  We study the asymptotics for large time of solutions to a one dimensional
parabolic evolution equation with non-standard measure-valued right hand side,
that involves derivatives of the solution computed at a free boundary point.
The problem is a particular case of a mean-field free boundary model proposed
by Lasry-Lions on price formation and dynamic equilibria.
  The main step in the proof is based on the fact that the free boundary
disappears in the linearized problem, thus can be treated as a perturbation
through semigroup theory. This requires a delicate choice for the function
spaces since higher regularity is needed near the free boundary. We show global
existence for solutions with initial data in a small neighborhood of any
equilibrium point, and exponential decay towards a stationary state. Moreover,
the family of equilibria of the equation is stable, as follows from center
manifold theory.
