  In this paper we consider a free boundary tumor growth model with a time
delay in cell proliferation and study how time delay affects the stability and
the size of the tumor. The model is a coupled system of an elliptic equation, a
parabolic equation and an ordinary differential equation. It incorporates the
cell location under the presence of time delay, with the tumor boundary as a
free boundary. A parameter $\mu$ in the model is proportional to the
"aggressiveness" of the tumor. It is proved that there exists a unique
classical radially symmetric stationary solution $(\sigma_*, p_*, R_*)$ which
is stable for any $\mu > 0$ with respect to all radially symmetric
perturbations (c.f. \cite{delay1}). However, under non-radially symmetric
perturbations, we prove that there exists a critical value $\mu_*$, such that
if $\mu<\mu_*$ then the stationary solution $(\sigma_*, p_*, R_*)$ is linearly
stable; whereas if $\mu>\mu_*$ the stationary solution is unstable. It is
actually unrealistic to expect the problem to be stable for large tumor
aggressiveness parameter, therefore our result is more reasonable. Furthermore,
we established that adding the time delay in the model would result in a larger
stationary tumor, and if the tumor aggressiveness parameter is larger, then the
time delay would have a greater impact on the size of the tumor.
