  We study a free boundary problem modeling multi-layer tumor growth with a
small time delay $\tau$, representing the time needed for the cell to complete
the replication process. The model consists of two elliptic equations which
describe the concentration of nutrient and the tumor tissue pressure,
respectively, an ordinary differential equation describing the cell location
characterizing the time delay and a partial differential equation for the free
boundary. In this paper we establish the well-posedness of the problem, namely,
first we prove that there exists a unique flat stationary solution $(\sigma_*,
p_*, \rho_*, \xi_* )$ for all $\mu>0$. The stability of this stationary
solution should depend on the tumor aggressiveness constant $\mu$. It is also
unrealistic to expect the perturbation to be flat. We show that, under non-flat
perturbations, there exists a threshold $\mu_*>0$ such that $(\sigma_*, p_*,
\rho_*, \xi_*)$ is linearly stable if $\mu<\mu_*$ and linearly unstable if
$\mu>\mu_*$. Furthermore, the time delay increases the stationary tumor size.
These are interesting results with mathematical and biological implications.
