Analysis of a free boundary problem for tumor growth with angiogenesis and time delays in proliferation

Abstract

In this paper we consider a free boundary problem for tumor growth with angiogenesis and time delays in the process of proliferation. The model is established by using reaction–diffusion dynamics and taking a time delay into account. In order to get more nutrients the tumor will attract blood vessels. Assume α(t) is the rate at which the tumor attracts blood vessels, so that ∂σ∂r+α(t)(σ−σ̄)=0 holds on the boundary, where σ̄ is the concentration of nutrients externally supplied to the tumor. When α is a constant, the stability of the unique stationary solution is proved. When α depends on time, we show that (i) R(t) will remain bounded if α(t) is bounded; (ii) limt→∞R(t)=0 if limt→∞α(t)=0; (iii) if α(t) is almost periodic and the nutrients supply outside the tumor is sufficient, there exists an almost periodic R(t).