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

## TL;DR

This paper proves convergence of a delayed heat equation with space-dependent operators and establishes global existence for a nonlinear adhesion model, advancing mathematical understanding of transient elastic linkages and adhesion forces.

## Contribution

It introduces new convergence results for space-dependent delay operators and extends fixed-point methods to prove global existence in a nonlinear adhesion model.

## Key findings

- Convergence of delay operators to friction terms with spatial dependence.
- Global existence of solutions prevents rupture scenarios.
- Extension of previous models to include space-dependent adhesion forces.

## Abstract

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.