Analysis of a tumor model free boundary problem with action of an inhibitor and nonlinear boundary conditions

Abstract

In this paper we study a free boundary problem modeling the growth of a tumor under the action of an inhibitor. Unlike other tumor models in the existing literature, the model under this study contains nonlinear boundary conditions which induces many new difficulties in rigorous mathematical analysis. Under the assumption that the tumor is spherically symmetric, we first establish global well-posedness of this model by proving that it admits a unique global classical solution. Next we study large-time behavior of the solution. We prove that the tumor cannot expand unboundedly, and give sufficient conditions to guarantee the tumor finally vanishing or persisting. We also give sufficient conditions to ensure that the model has at least one stationary solution (σs(r),βs(r),Rs), and study asymptotic stability of the stationary solution.