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

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.