We will present some recent results concerning the long-time dynamics of reaction di usion equations with obstacle potentials. For these kind of systems, less is known regarding the finite dimensional reduction for the long-time dynamics, namely, the finite dimensionality in terms of fractal dimension of the global attractors. The main obstruction is caused by the singular character of the potential involved that makes the solution operator non-di erentiable with respect to
initial data. As a consequence, the standard method of volume contractions can not be applied. By making use of proper modi cations of the method of "l-trajectories", we will show that when the obstacle K is a closed, convex and bounded subset of R^n with smooth boundary or it is a closed n-dimensional simplex, the long-time behavior of the solution semigroup associated with the problem can be described in terms of an exponential attractor, which has fi nite fractal dimension. Thus, the global attractor, being contained in the exponential one, will have finite fractal dimension too. The core of the proof relies on a careful (and non-standard) approximation argument and on some (apparently) new estimates on the
solutions. This is a joint work with S. Zelik (Surrey).