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June  2019, 12(3): 551-571. doi: 10.3934/krm.2019022

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

 1 Laboratoire de Géodésie, IGN-LAREG, Bâtiment Lamarck A et B, 35 rue Hélène Brion, 75013 Paris, France 2 Sorbonne Universités, Inria, UPMC Univ Paris 06, Mamba project-team, Laboratoire Jacques-Louis Lions, Paris, France 3 Wolfgang Pauli Institute, c/o Faculty of Mathematics of the University of Vienna, Vienna, Austria 4 Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 45 Avenue des États-Unis, 78035 Versailles cedex, France

* Corresponding author: Marie Doumic

Received  January 2018 Revised  June 2018 Published  February 2019

Fund Project: M.D. is supported by ERC Starting Grant SKIPPERAD (number 306321).
P.G. is supported by ANR project KIBORD, ANR-13-BS01-0004.

We study the asymptotic behaviour of the following linear growth-fragmentation equation
 $\frac{\partial}{\partial t} u(t,x) + \dfrac{\partial}{ \partial x} \big(x u(t,x)\big) + B(x) u(t,x) = 4 B(2x)u(t,2x),$
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 hypocoercivity 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-diffusive numerical scheme, able to capture the oscillations.
Citation: Étienne Bernard, Marie Doumic, Pierre Gabriel. Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts. Kinetic & Related Models, 2019, 12 (3) : 551-571. doi: 10.3934/krm.2019022
##### References:

show all references

##### References:
The real part for the three first eigenvectors ${\mathcal U} _0,\, {\mathcal U} _1,\, {\mathcal U} _2$ for $B(x) = x^2$. We see the oscillatory behaviour for ${\mathcal U} _1$ and ${\mathcal U} _2$
Two different initial conditions

Left: peak in $x = 2.$ Right: $u^{\rm{in}} (x) = x^2\exp(-x^2/2)$.

Time evolution of $\max\limits_{x>0} u(t,x)e^{-t}$

Left: for the peak as initial condition. Right: for the smooth initial condition.

Size distribution $u(t,x)e^{-t}$ at five different times (each time is in a different grey). Left: for the peak as initial condition. Right: for the smooth initial condition
Left: initial distribution (full blue line) and dominant eigenvector (doted red line), for $B(x) = x^3$. We see that the constant such that $u^{\rm{in}}\leq {\mathcal U}_0$ is very large. Right: time evolution of Error$_{E_2^n}$ (doted red line) and Error Mean$_{E_2^n}$ (full blue line), in a log scale for the ordinates
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