Analysis of a high-dimensional free boundary problem on tumor growth with time-dependent nutrient supply and inhibitor action

Abstract

This paper is concerned with a free boundary problem modeling tumor growth with time-dependent nutrient supply and inhibitor action. We highlight in this paper that the spatial domain occupied by the tumor is set to be n-dimensional for any $n⩾3$, and it is taken into account that the nutrient supply $\varphi \left(t\right)$ and the inhibitor injection $\psi \left(t\right)$ on the tumor surface are time-varying in this problem. The high-dimensional setting of the problem makes the proof of the existence of radial stationary solutions and the accurate determination of their numbers highly nontrivial, in which we have developed a new method that is different from the previous work by Cui and Friedman [11]. We can give a complete classification of the radial stationary solutions to this problem under different parameter conditions, and also explore the asymptotic behavior of the transient solution for small $c:=c_1+c_2$ in the case that $\varphi \left(t\right)$ and $\psi \left(t\right)$ have finite limits as $t\to \infty$.