Linear stability for a free boundary tumor model with a periodic supply of external nutrients

In this paper, we consider a free boundary tumor model with a periodic supply of external nutrients, so that the nutrient concentration sigma satisfies sigma = phi(t) on the boundary, where phi(t) is a positive periodic function with period T. A parameter mu in the model is proportional to the "aggressiveness" of the tumor. If 0<sigma similar to<min0 <= t <= T phi(t), where sigma similar to is a threshold concentration for proliferation, Bai and Xu [Pac J Appl Math. 2013;5;217-223] proved that there exists a unique radially symmetric T-periodic positive solution (sigma(*)(r,t),p(*)(r,t),R-*(t)), which is stable for any mu > 0 with respect to all radially symmetric perturbations. We prove that under nonradially symmetric perturbations, there exists a number mu(*) such that if 0 < mu < mu(*), then the T-periodic solution is linearly stable, whereas if mu > mu(*), then the T-periodic solution is linearly unstable.