  Several results in previous works, strongly depend on the exponential tail of
the linkages' distribution in our adhesive models. The purpose of this paper is
to weaken this hypothesis and to allow more fat tails for large ages. From the
biological point of view this means that we allow adhesions to be stronger,
because linkages break less often. Moreover, in our previous articles, the
asymptotic expansion of adhesion site's position and the corresponding error
estimates also used some fast decay properties of the kernel, we show, when the
kernel is a given function of age but constant in time, how to overcome this
problem and construct asymptotic expansions in a systematic way at any order
with respect to a small parameter $\epsilon$ representing the linkages'
turnover.
