Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation

Abstract

We are interested in a non-local partial differential equation modeling equal mitosis. We prove that the solutions present persistent asymptotic oscillations and that the convergence to this periodic behavior, in suitable spaces of weighted signed measures, occurs exponentially fast. It can be seen as a spectral gap result between the countable set of dominant eigenvalues and the rest of the spectrum, which is to our knowledge completely new. The two main difficulties in the proof are to define the projection onto the subspace of periodic (rescaled) solutions and to estimate the speed of convergence to this projection. The first one is addressed by using the generalized relative entropy structure of the dual equation, and the second is tackled by applying Harris's ergodic theorem on sub-problems.

Asymptotic behaviors of a kinetic approach to the collective dynamics of a rock-paper-scissors binary game

“…A detailed proof for a similar result is given in [7] for weigthed measures and can be adapted to the present case. Yet we provide the main elements of the proof.…”

“…The large time asymptotics when h remains fixed was yet to be investigated: so is the purpose of the present paper. Our goal is to provide a precise asymptotic behavior to this equation, drawing inspiration from the methodology developped in [7] for a critical case of the growth-fragentation equation. In this article, the authors worked in a measure framework and adopted a combination of semigroup and duality approach.…”

Confining integro-differential equations originating from evolutionary biology: Ground states and long time dynamics

“…However, if we had u α ≡ 0, thanks to Lemma 3.1 we would have u α > 0 in R N and there would exist ε 0 > 0 small enough such that u α+ε0 := u α − ε 0 ψ > 0 on the ball B(0, R). Proceeding as in the proof of ( 20), we would deduce that u α+ε0 ≥ 0 in R N and thus α + ε 0 ∈ A, the set defined in (21), contradicting the definition of α. Therefore we must have u α ≡ 0, that is ϕ = αψ.…”

“…2. Since the work of S. Mischler and J. Scher [29] in 2016, quantifying the spectral gap of non-local and non-conservative linear equations is an active field of research, see [5,12,14,15,21]. To our knowledge, the result in Theorem 2.4 is the first quantified spectral gap result in the literature for Equation (3).…”

A non‐conservative Harris ergodic theorem

“…We give here the details of the construction of the growth-fragmentation semigroup and prove its basic properties, along the lines of [44,52,53]. We first prove that this indeed defines uniquely the family (𝑀 𝑡 𝑓) 𝑡⩾0 .…”