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

From transient elastic linkages to friction: a complete study of a penalized fourth order equation with delay