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

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