Qualitative analysis of a time-delayed free boundary problem for tumor growth with angiogenesis and Gibbs-Thomson relation

Abstract

In this paper we consider a time-delayed mathematical model describing tumor growth with angiogenesis and Gibbs-Thomson relation. In the model there are two unknown functions: One is σ(r, t) which is the nutrient concentration at time t and radius r, and the other one is R(t) which is the outer tumor radius at time t. Since R(t) is unknown and varies with time, this problem has a free boundary. Assume α(t) is the rate at which the tumor attracts blood vessels and the Gibbs-Thomson relation is considered for the concentration of nutrient at outer boundary of the tumor, so that on the outer boundary, the condition ∂σ ∂r + α(t) (σ − N(t)) = 0, r = R(t) holds, where N(t) = σ¯ 1 − Rγ(t)! H(R(t)) is derived from Gibbs-Thomson relation. H(·) is smooth on (0, ∞) satisfying H(x) = 0 if x ≤ γ, H(x) = 1 if x ≥ 2γ and 0 ≤ H0(x) ≤ 2/γ for all x ≥ 0. In the case where α is a constant, the existence of steady-state solutions is discussed and the stability of the steady-state solutions is proved. In another case where α depends on time, we show that R(t) will be also bounded if α(t) is bounded and some sufficient conditions for the disappearance of tumors are given.