The existence and linear stability of periodic solution for a free boundary problem modeling tumor growth with a periodic supply of external nutrients. (August 2021) - He, Wenhua; Xing, Ruixiang
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 ˜ ⩾ 1 T ∫ 0 T Φ ( t ) d t ; (ii) If σ ˜ < 1 T ∫ 0 T Φ ( t ) d t, 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 ∗ .