Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation

In this paper, a free boundary problem modeling tumor growth under the direct effect of an inhibitor with time delays is studied. The delays represent the time taken for cells to undergo mitosis. Nonnegativity of solutions, the existence of the stationary solutions and their asymptomatic behavior are studied. The results show that when the inhibitor is large, and the initial tumor is not too large, the tumor will disappear. If however, the initial tumor is large enough, then it will grow. When the inhibitor is not as large, the growth of the tumor is determined by the size of the nutrients and whether the initial tumor is large or not. When the inhibitor is smaller, the tumor will grow no matter if the initial tumor is large or not.