Analysis of the Solution to a Labelled Cell Populations System

In this paper, we study the solution to a labelled cell population system with variable coefficients and then investigate the expansion of the solution to the system. Using the spectral analysis technique, we show that the spectrum consists of all eigenvalues of the operator A determined by the system and distributes in a strip parallel to the imaginary axis. In particular, we give asymptotic expression of all eigenvalues. Furthermore, we prove that there is a sequence of generalized eigenfunctions of A that forms a Riesz basis for the Hilbert space. Hence we can expand the solution of the system by using its eigenfunctions and assert that the system associates a C₀-group.