  We consider a linear integro-differential equation which arises to describe
both aggregation-fragmentation processes and cell division. We prove the
existence of a solution $(\lb,\U,\phi)$ to the related eigenproblem. Such
eigenelements are useful to study the long time asymptotic behaviour of
solutions as well as the steady states when the equation is coupled with an
ODE. Our study concerns a non-constant transport term that can vanish at $x=0,$
since it seems to be relevant to describe some biological processes like
proteins aggregation. Non lower-bounded transport terms bring difficulties to
find $a\ priori$ estimates. All the work of this paper is to solve this problem
using weighted-norms.

Analysis of PDEs