  In this paper we study the asymptotic behaviour of a nonlocal nonlinear
parabolic equation governed by a parameter. After giving the existence of
unique branch of solutions composed by stable solutions in stationary case, we
gives for the parabolic problem $L^\infty $ estimates of solution based on
using the Moser iterations and existence of global attractor. We finish our
study by the issue of asymptotic behaviour in some cases when $t\to \infty$.
