Summary: In this paper, we present some existence results on \(L^{1}\) spaces of a nonlinear boundary value problem derived from a model introduced by M. Rotenberg [“Transport theory for growing cell populations”, J. Theor. Biol. 103, No. 2, 181–199 (1983; doi:10.1016/0022-5193(83)90024-3)] describing the growth of a cell population. Each cell of this population is distinguished by its degree of maturity \(\mu \in [0,1]\) and its maturation velocity \(v\). The biological boundary at \(\mu = 0\) and \(\mu = 1\) are fixed and tightly coupled through the mitosis. At mitosis, daughter cells and mother cells are related by a general reproduction rule, which covers all known biological ones. In this work, the maturation velocity is allowed to be infinite, that is, \(v \in [0, +\infty)\). This hypothesis introduce some mathematical difficulties, which are overcomed by using a measure of weak noncompactness adapted to the problem and a recent fixed point theorem (Theorem 3.2) involving weakly compact operators on nonreflexive Banach spaces.