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 $\Phi (t)$ of external nutrients. The model contains two parameters $\mu$ and $\tilde{\sigma }$. We first show that (i) zero radially symmetric solution is globally stable if and only if $\tilde{\sigma }\ge \frac{1}{T}{\int }_0^T\Phi (t)dt$; (ii) If $\tilde{\sigma }<\frac{1}{T}{\int }_0^T\Phi (t)dt$, then there exists a unique radially symmetric positive solution $\left({\sigma }_{\ast }(r,t),p_{\ast }(r,t),R_{\ast }(t)\right)$ with period $T$ and it is a global attractor of all positive radially symmetric solutions for all $\mu >0$. 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 ${\mu }_{\ast }>0$ such that $\left({\sigma }_{\ast }(r,t),p_{\ast }(r,t),R_{\ast }(t)\right)$ is linearly stable for $\mu <{\mu }_{\ast }$ and linearly unstable for $\mu >{\mu }_{\ast }$.