  In this paper we consider a fourth order nonlinear parabolic delayed problem
modelling a quasi-instantaneous turn-over of linkages in the context of
cell-motility. The model depends on a small parameter $\epsilon$ which
represents a typical time scale of the memory effect. We first prove global
existence and uniqueness of solutions for $\epsilon$ fixed. This is achieved by
combining suitable fixed-point and energy arguments and by uncovering a
nonlocal in time, integral conserved quantity. After giving a complete
classification of steady states in terms of elliptic functions, we next show
that every solution converges to a steady state as $t \to \infty$. When
$\epsilon \to 0$, we then establish convergence results on finite time
intervals, showing that the solution tends in a suitable sense towards the
solution of a parabolic problem without delay. Moreover, we establish the
convergence of energies as $\epsilon \to 0$, which enables us to show that, for
$\epsilon$ small enough, the $\epsilon$-dependent problem inherits part of the
large time asymptotics of the limiting parabolic problem.
