  We study the relaxation times for a parabolic differential equation whose
solution represents the atom dislocation in a crystal. The equation that we
consider comprises the classical Peierls-Nabarro model as a particular case,
and it allows also long range interactions.
  It is known that the dislocation function of such a model has the tendency to
concentrate at single points, which evolve in time according to the external
stress and a singular, long range potential.
  Depending on the orientation of the dislocation function at these points, the
potential may be either attractive or repulsive, hence collisions may occur in
the latter case and, at the collision time, the dislocation function does not
disappear.
  The goal of this paper is to provide accurate estimates on the relaxation
times of the system after collision. More precisely, we take into account the
case of two and three colliding points, and we show that, after a small
transition time subsequent to the collision, the dislocation function relaxes
exponentially fast to a steady state.
  We stress that the exponential decay is somehow exceptional in nonlocal
problems (for instance, the spatial decay in this case is polynomial). The
exponential time decay is due to the coupling (in a suitable space/time scale)
between the evolution term and the potential induced by the periodicity of the
crystal.

Analysis of PDEs