Lord Kelvin's method of images approach to the Rotenberg model and its asymptotics

We study a mathematical model of cell populations dynamics proposed by M. Rotenberg and investigated by M. Boulanouar. Here, a cell is characterized by her maturity and speed of maturation. The growth of cell populations is described by a partial differential equation with a boundary condition. In the first part of the paper we exploit semigroup theory approach and apply Lord Kelvin's method of images in order to give a new proof that the model is well posed. Next, we use a semi-explicit formula for the semigroup related to the model obtained by the method of images in order to give growth estimates for the semigroup. The main part of the paper is devoted to the asymptotic behaviour of the semigroup. We formulate conditions for the asymptotic stability of the semigroup in the case in which the average number of viable daughters per mitosis equals one. To this end we use methods developed by K. Pich\'or and R. Rudnicki.