Linear stability for a periodic tumor angiogenesis model with free boundary in the presence of inhibitors

Abstract

In this paper, we research the issue of the free boundary of vascularized tumor growth using a T-periodic supply ψ(t) of outside nutrients and inhibitors. The model consists of two reaction diffusion equations, an elliptic equation and an ordinary differential equation. The reaction diffusion equations describe the nutrient and inhibitor concentrations. The internal pressure distribution is described by the elliptic equation. The ODE describes the boundary value condition of the tumor model. After some meticulous mathematical analysis of the model system, we prove the existence and uniqueness of the radially symmetric T-periodic positive solution with u˜≤min0≤t≤T⁡ψ(t), where u˜ is a parameter, denoting a threshold concentration for proliferation. Next, we further demonstrate the existence of a μ⁎>0 such that (u⁎(r,t),v⁎(r,t),p⁎(r,t),R⁎(t)) is linearly stable for μ<μ⁎ and linearly unstable for μ>μ⁎ under perturbations that are not radially symmetric, where μ is a constant, representing the “intensity” of mitosis-induced cell growth.