Title: Lord Kelvin's method of images approach to the Rotenberg model and its asymptotics,
Abstract: 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.
Pichór and R. Rudnicki.