Abstract:

In this paper, we consider a free boundary problem modeling the growth of tumors with angiogenesis and a $\omega$-periodic supply of external nutrients $\phi(t)$. Denote by $S(\sigma)$ the proliferation rate of tumor cells. We first establish the well-posedness and then give a complete classification of asymptotic behavior of solutions according to the sign of $\frac1{\omega}\int_0^\omega S(\phi(t)){\rm d}t$. It is shown that if $\frac1{\omega}\int_0^\omega S(\phi(t)){\rm d}t\le0$, then all evolutionary tumors will finally vanish; the converse is also true. If instead $\frac1{\omega}\int_0^\omega S(\phi(t)){\rm d}t>0$, then there exists a unique and stable positive periodic solution.

Key words: Free boundary problem, Necrotic tumor, Angiogenesis, Periodic solution, Stability