Linear stability for a periodic tumor angiogenesis model with free boundary

In this paper, we consider an angiogenesis free boundary tumor model with external periodic nutrient supply. The model is in the form of a reaction-diffusion equation describing the concentration of nutrients sigma and an elliptic equation describing the distribution of the internal pressure p. The vasculature provides a periodic supply of nutrients to the tumor at a rate proportional to beta, so that partial derivative sigma/partial derivative n+beta(sigma-phi(t)) = 0 holds on the boundary, where phi(t) is the nutrient concentration outside the tumor. Here phi(t) is a periodic function with period T and satisfies phi(t) = phi(t + T). A parameter mu in the model expresses the "aggressiveness" of the tumor. We prove that under non-radially symmetric perturbations, there exists mu(*) > 0 such that the T-periodic solution is linearly stable for mu < mu(*), and is linearly unstable for mu > mu(*). (C) 2020 Elsevier Ltd. All rights reserved.