On an Unbounded Linear Operator Arising in the Theory of Growing Cell Population

This paper deals with the spectral analysis of a class of unbounded linear operators corresponding to a partial differential equation originally proposed by J. L. Lebowitz and S. I. Rubinow J. Math. Biol. 1, 1974, 17]36. to model an age structured proliferating cell population. Individual cells are distinguished by age and cell cycle length. The cell cycle length is considered as an inherited property determined at birth. After a detailed spectral analysis for general boundary conditions which model the process of cell division of mother cell and the inherence of cycle length by daughter cells. it is shown that the associated Cauchy problem is governed by a C0-semigroup. A spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given.