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Journal Articles Kinetic and Related Models Year : 2019

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.
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Dates and versions

hal-01363549 , version 1 (09-09-2016)
hal-01363549 , version 2 (31-01-2017)
hal-01363549 , version 3 (01-11-2017)
hal-01363549 , version 4 (16-01-2018)
hal-01363549 , version 5 (17-11-2018)

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Cite

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

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