Analysis of non-overlapping models with a weighted infinite delay

The framework of this article is cell motility modeling.

Approximating cells as rigid spheres we take into account for both non-penetration and adhesions forces.

Adhesions are modeled as a memory-like microscopic elastic forces.

This leads to a delayed and constrained vector valued system of equations.

We prove that the solution of these equations converges when {\epsilon}, the linkages turnover parameter, tends to zero to the a constrained model with friction.

We discretize the problem and penalize the constraints to get an uncon?strained minimization problem.

The well-posedness of the constrained problem is obtained by letting the penalty parameter to tend to zero.

Energy estimates `a la De Giorgi are derived accounting for delay.

Thanks to these estimates and the convexity of the constraints, we obtain compactness uniformly with respect to the discretisation step and {\epsilon}, this is the mathematically involved part of the article.

Considering that the characteristic bonds lifetime goes to zero, we recover a friction model comparable to [Venel et al, ESAIM, 2011] but under more realistic assumptions on the external load, this part being also one of the challenging aspects of the work