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

TL;DR

This paper analyzes the long-term behavior of solutions to a non-local PDE modeling equal mitosis, showing they exhibit persistent oscillations and converge exponentially fast to a periodic state due to a spectral gap.

Contribution

It establishes the existence of persistent oscillations and exponential convergence in measure solutions to an equal mitosis equation, introducing a novel spectral gap analysis.

Findings

Solutions exhibit persistent asymptotic oscillations.

Convergence to periodic solutions occurs exponentially fast.

Spectral gap between dominant eigenvalues and the rest of the spectrum is demonstrated.

Abstract

We are interested in a non-local partial differential equation modeling equal mitosis. We prove that the solutions present persistent asymptoticoscillations and that the convergence to this periodic behavior, in suitable spaces of weighted signed measures, occurs exponentially fast. It can beseen as a result of spectral gap 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 thespeed of convergence to this projection. The first one is addressed by using the generalized relative entropy structure of the dual equation, and thesecond is tackled by applying Harris's ergodic theorem on sub-problems.