Analysis of a free-boundary tumor model with angiogenesis

We consider a free boundary problem for a spherically symmetric tumor with free boundary r < R(t). In order to receive nutrients u the tumor attracts blood vessel at a rate proportional to alpha(t), so that partial derivative u/partial derivative r + alpha(t)(u - <(u)over bar>) = 0 holds on the boundary, where (u) over bar is the nutrient concentration outside the tumor. A parameter mu in the model is proportional to the 'aggressiveness' of the tumor. When alpha is a constant, the existence and uniqueness of stationary solution is proved. For the more general situation when alpha depends on time, we show, under various conditions (that are always satisfied if mu is small), that (i) R(t) remains bounded if alpha(t) remains bounded; (ii) lim(t ->infinity) R(t) = 0 if lim(t ->infinity) alpha(t) = 0; and (iii) lim inf(t ->infinity) R(t) > 0 if lim inf(t ->infinity) alpha(t) > 0. Surprisingly, we exhibit solutions (when mu is not small) where alpha(t) -> 0 exponentially in t while R(t) -> infinity exponentially in t. Finally, we prove the global asymptotic stability of steady state when mu is sufficiently small. (C) 2015 Elsevier Inc. All rights reserved.