Eigenelements of a general aggregation-fragmentation model

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 (λ, $\mathcal U$, φ) to the related eigenproblem. Such eigenelements are useful to study the long-time asymptotic behavior 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.