The existence and linear stability of periodic solution for a free boundary problem modeling tumor growth with a periodic supply of external nutrients

Abstract

We study a free boundary problem modeling tumor growth with a T-periodic supply Φ(t) of external nutrients. The model contains two parameters μ and σ˜. We first show that (i) zero radially symmetric solution is globally stable if and only if σ˜⩾1T∫0TΦ(t)dt; (ii) If σ˜<1T∫0TΦ(t)dt, then there exists a unique radially symmetric positive solution σ∗(r,t),p∗(r,t),R∗(t) with period T and it is a global attractor of all positive radially symmetric solutions for all μ>0. These results are a perfect answer to open problems in Bai and Xu [Pac. J. Appl. Math. 2013(5), 217–223]. Then, considering non-radially symmetric perturbations, we prove that there exists a constant μ∗>0 such that σ∗(r,t),p∗(r,t),R∗(t) is linearly stable for μ<μ∗ and linearly unstable for μ>μ∗.