Eigenelements of a General Aggregation-Fragmentation Model

Marie Doumic-Jauffret (1) , Pierre Gabriel (2)

1 BANG - Nonlinear Analysis for Biology and Geophysical flows
2 LJLL - Laboratoire Jacques-Louis Lions

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 $(\lambda,\mathcal U,\phi)$ 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.

Mots clés: size repartition; eigenproblem; Aggregation-fragmentation equations; long-time asymptotic; cell division; polymerization process

Domaines: Equations aux dérivées partielles [math.AP]

Marie Doumic-Jauffret, Pierre Gabriel. Eigenelements of a General Aggregation-Fragmentation Model. Mathematical Models and Methods in Applied Sciences, 2010, 20 (5), pp.757-783. ⟨10.1142/S021820251000443X⟩. ⟨hal-00408088v3⟩