Some remarks on the solution of the cell growth equation

Abstract

A process of growth and division of cells is modelled by an initial boundary value problem that involves a first-order linear functional partial differential equation, the so-called sell growth equation. The analytical solution to this problem was given in the paper Zaidi et al. (Zaidi et al. 2015 Solutions to an advanced functional partial differential equation of the pantograph type (Proc. R. Soc. A 471, 20140947 (doi:10.1098/rspa.2014.0947)). In this note, we simplify the arguments given in the paper mentioned above by using the theory of operator semigroups. This theory enables us to prove the existence and uniqueness of the solution and to express this solution in terms of Dyson–Phillips series. The asymptotics of the solution is also discussed from the point of view of the theory of operator semigroups.