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

Abstract

We study the asymptotic behaviour of the following linear growthfragmentation equation ∂/ ∂t u(t,x) + ∂/ ∂x (xu(t,x)+ B(x)u(t,x) = 4B(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 L2 space, where well-posedness is obtained via semigroup analysis. We also propose a non-diffusive numerical scheme, able to capture the oscillations.