No abstract

In this paper we consider a free boundary tumor model with angiogenesis. The model consists of a reaction-diffusion equation describing the concentration of nutrients σ and an elliptic equation describing the distribution of the internal pressure p. The vasculature supplies nutrients to the tumor, so that ∂σ ∂n + β(σ −σ) = 0 holds on the boundary, where a positive constant β is the rate of nutrient supply to the tumor andσ is the nutrient concentration outside the tumor. The tumor cells proliferate at a rate µ. If 0 < σ < σ, where σ is a threshold concentration for proliferating, then there exists a unique radially symmetric stationary solution (σ S (r), p S (r), R S). In this paper, we found a function µ * = µ * (R S) such that if µ < µ * then the radially symmetric stationary solution is linearly stable with respect to non-radially symmetric perturbations, whereas if µ > µ * then the radially symmetric stationary solution is linearly unstable.

In this paper, we consider a free boundary tumor model with a periodic supply of external nutrients, so that the nutrient concentration satisfies = (t) on the boundary, where (t) is a positive periodic function with period T. A parameter in the model is proportional to the "aggressiveness" of the tumor.If 0 <̃< min 0≤t≤T (t), wherẽis 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 ( * (r, t), p * (r, t), R * (t)), which is stable for any > 0 with respect to all radially symmetric perturbations. 17 We prove that under nonradially symmetric perturbations, there exists a number * such that if 0 < < * , then the T-periodic solution is linearly stable, whereas if > * , then the T-periodic solution is linearly unstable.

No abstract