  In the first part of this work we show the convergence with respect to an
asymptotic parameter {\epsilon} of a delayed heat equation. It represents a
mathematical extension of works considered previously by the authors [Milisic
et al. 2011, Milisic et al. 2016]. Namely, this is the first result involving
delay operators approximating protein linkages coupled with a spatial elliptic
second order operator. For the sake of simplicity we choose the Laplace
operator, although more general results could be derived. The main arguments
are (i) new energy estimates and (ii) a stability result extended from the
previous work to this more involved context. They allow to prove convergence of
the delay operator to a friction term together with the Laplace operator in the
same asymptotic regime considered without the space dependence in [Milisic et
al, 2011]. In a second part we extend fixed-point results for the fully
non-linear model introduced in [Milisic et al, 2016] and prove global existence
in time. This shows that the blow-up scenario observed previously does not
occur. Since the latter result was interpreted as a rupture of adhesion forces,
we discuss the possibility of bond breaking both from the analytic and
numerical point of view.
