  In this paper, we consider a free boundary problem modeling the growth of
spherically symmetric necrotic tumors with angiogenesis and a $\omega$-periodic
supply $\phi(t)$ of external nutrients. In the model, the consumption rate of
the nutrient and the proliferation rate of tumor cells $S(\sigma)$ are both
general nonlinear functions. The well-posedness and asymptotic behavior of
solutions are studied. We show that if the average of $S(\phi(t))$ is
nonpositive, then all evolutionary tumors will finally vanish; the converse is
also ture. If instead the average of $S(\phi(t))$ is positive, then there
exists a unique positive periodic solution and all other evolutionary tumors
will converge to this periodic state.
