Eigenelements of a General Aggregation-Fragmentation Model

TL;DR

This paper analyzes a linear integro-differential equation modeling aggregation, fragmentation, and cell division, proving the existence of eigen-elements to understand long-term behavior, especially with complex transport terms relevant to biological processes.

Contribution

It establishes the existence of eigen-elements for a complex integro-differential equation with non-constant, potentially vanishing transport terms, using weighted-norms.

Findings

Existence of eigen-elements for the model.

Handling of non-lower-bounded transport terms.

Application to biological aggregation processes.

Abstract

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.