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 σ satisfies σ  =  ϕ ( t ) on the boundary, where ϕ ( t ) is a positive periodic function with period T . A parameter μ in the model is proportional to the “aggressiveness” of the tumor. If 0 < σ ˜ < min 0 ≤ t ≤ T ϕ ( t ) , where σ ˜ 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 ( σ ∗ ( r , t ), p ∗ ( r , t ), R ∗ ( t )), which is stable for any μ  > 0 with respect to all radially symmetric perturbations.[17][Bai M, 2013] We prove that under nonradially symmetric perturbations, there exists a number μ ∗ such that if 0 <  μ  <  μ ∗ , then the T ‐periodic solution is linearly stable, whereas if μ  >  μ ∗ , then the T ‐periodic solution is linearly unstable.