This paper investigates the Rotenberg equation with non-smooth boundary conditions in L¹ space. We prove that the main operator of the equation is densely defined and resolvent-positive. Furthermore, it is shown that the positive cone in the domain of its adjoint operator is cofinal within the positive cone of the entire adjoint space. Consequently, the spectral bound of the semigroup generated by the main operator is demonstrated to coincide with its growth order. Additionally, the continuous dependence of solutions on boundary parameters is rigorously established.

In 1983, Rotenberg firstly proposed the Rotenberg model, which has attracted wide attention. For Rotenberg model, many previous studies have been carried out, but the time delay of cell division was not considered in previous studies. In order to describe cell proliferation better and more accurately, on the basis of previous studies, we consider the time delay of cell division, make some modifications to the original Rotenberg model, and prove that transport operator A_H can generate a C₀ semigroup. The spectrum of the corresponding transport operator is analyzed, and it is only consists of finite discrete eigenvalues with finite algebraic multiplicity in the region $\mathrm{\Gamma }=\sigma \left({\mathrm{A}}_{\mathrm{H}}\right)\cap \{\lambda \in C\mid \mathrm{Re}\lambda >\gamma \}\left(\gamma >{\lambda }_0\right)$.