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2019, Volume 12, Issue 3: 551-571. Doi: 10.3934/krm.2019022

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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
^* Corresponding author: Marie Doumic

Received: January 2018

Revised: June 2018

Published: June 2019

M.D. is supported by ERC Starting Grant SKIPPER^AD (number 306321).
P.G. is supported by ANR project KIBORD, ANR-13-BS01-0004

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35Q92, 35B10, 35B40, 47D06, 35P05; Secondary: 35B41, 92D25, 92B25.

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Figure 1. 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 $

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Figure 2. Two different initial conditions

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Figure 3. Time evolution of $ \max\limits_{x>0} u(t,x)e^{-t} $

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Figure 4. 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

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Figure 5. 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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