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In the first part of this work we show the convergence with respect to anasymptotic parameter {\epsilon} of a delayed heat equation. It represents amathematical extension of works considered previously by the authors [Milisicet al. 2011, Milisic et al. 2016]. Namely, this is the first result involvingdelay operators approximating protein linkages coupled with a spatial ellipticsecond order operator. For the sake of simplicity we choose the Laplaceoperator, although more general results could be derived. The main argumentsare (i) new energy estimates and (ii) a stability result extended from theprevious work to this more involved context. They allow to prove convergence ofthe delay operator to a friction term together with the Laplace operator in thesame asymptotic regime considered without the space dependence in [Milisic etal, 2011]. In a second part we extend fixed-point results for the fullynon-linear model introduced in [Milisic et al, 2016] and prove global existencein time. This shows that the blow-up scenario observed previously does notoccur. Since the latter result was interpreted as a rupture of adhesion forces,we discuss the possibility of bond breaking both from the analytic andnumerical point of view.

Space dependent adhesion forces mediated by transient elastic linkages : new convergence and global existence results