Eigenelements of a General Aggregation-Fragmentation Model

By Marie Doumic-Jauffret, Pierre Gabriel. 2009

122 citations recorded by the source corpus.

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, , 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.

Subfields: Mathematical Physics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/f112e4350ede92336652ec7b69b9a025c2671a07

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Citations in the corpus

A non‐conservative Harris ergodic theorem (2019, background): “The eigenvalue problem has been solved in [56, 31] for general coefficients.”
Statistical estimation of a growth-fragmentation model observed on a genealogical tree (2012, methodology): “By use of the general relative entropy principle, Michel, Mischler and Perthame showed that the approximation n ( t, x ) e − λ 0 t ≈ N ( x ) is valid [29], with λ 0 > 0, and where ( λ 0 , N ) is the dominant eigenpair related to the corresponding eigenvalue problem, see [28, 32, 12, 25, 8, 2].”
Limit theorems for some branching measure-valued processes (2011, background): “! In order to have the many-to-one formula, we give a condition for the existence of eigenelement extracted to [46] (see [45], for an asymmetric division cell, and [19], for a non linear motion between the division).”