Linear stability for a free boundary problem modeling the growth of tumor cord with time delay

This paper was concerned with a free boundary problem modeling the growth of tumor cord with a time delay in cell proliferation, in which the cell location was incorporated, the domain was bounded in $ \mathbb{R}^2 $, and its boundary included two disjoint closed curves, one fixed and the other moving and a priori unknown. A parameter $ \mu $ represents the aggressiveness of the tumor. We proved that there exists a unique radially symmetric stationary solution for sufficiently small time delay, and this stationary solution is linearly stable under the nonradially symmetric perturbations for any $ \mu > 0 $. Moreover, adding the time delay in the model leads to a larger stationary tumor. If the tumor aggressiveness parameter is bigger, the time delay has a greater effect on the size of the stationary tumor, but it has no effect on the stability of the stationary solution.