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 ϕ(t) and the inhibitor injection ψ(t) 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:=c1+c2 in the case that ϕ(t) and ψ(t) have finite limits as t→∞.
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