A Bohr-Neugebauer property for abstract almost periodic evolution equations in Banach spaces: Application to a size-structured population model

This work continues the investigations done in the literature about the so called Bohr-Neugebauer property for almost periodic differential equations in Hilbert spaces. We give a new sufficient condition to ensure that an almost periodic linear evolution equation in a Banach space has the Bohr-Neugebauer property. More specifically, we investigate when all the bounded solutions on R are almost periodic. This new sufficient condition involves the essential growth bound of the C0-semigroup generated by the autonomous part of the equation. This enables us to investigate the Bohr-Neugebauer property for partial differential equations with solutions living in purely Banach spaces. An application to a size-structured population model will be studied using the theory of positive C0-semigroups.