Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts

Abstract : We study the asymptotic behaviour of the following linear growth-fragmentation equation $$\dfrac{\partial}{\partial t} u(t,x) + \dfrac{\partial}{\partial x} \big(x u(t,x)\big) + B(x) u(t,x) =4 B(2x)u(t,2x),$$ and prove that under fairly general assumptions on the division rate $B(x),$ its solution converges towards an oscillatory function, explicitely given by the projection of the initial state on the space generated by the countable set of the dominant eigenvectors of the operator. Despite the lack of hypo-coercivity of the operator, the proof relies on a general relative entropy argument in a convenient weighted $L^2$ space, where well-posedness is obtained via semigroup analysis. We also propose a non-dissipative numerical scheme, able to capture the oscillations.
Document type :
Journal articles
Complete list of metadatas

https://hal.archives-ouvertes.fr/hal-01363549
Contributor : Marie Doumic <>
Submitted on : Saturday, November 17, 2018 - 6:49:35 AM
Last modification on : Wednesday, October 9, 2019 - 4:34:02 PM
Long-term archiving on : Monday, February 18, 2019 - 12:13:38 PM

Files

BernardDoumicGabriel_final_hal...
Files produced by the author(s)

Identifiers

Relations

Citation

Etienne Bernard, Marie Doumic, Pierre Gabriel. Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts. Kinetic and Related Models , AIMS, 2019, 12 (3), pp.551-571. ⟨10.3934/krm.2019022⟩. ⟨hal-01363549v5⟩

Share

Metrics

Record views

361

Files downloads

282