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

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 $\sigma(r,t)$ 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 $\alpha(t)$ 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 $$\dfrac{\partial \sigma}{\partial r}+\alpha(t)\left(\sigma-N(t)\right)=0,~~r=R(t)$$ holds, where $N(t)=\bar{\sigma}\left(1-\dfrac{\gamma}{R(t)}\right)H(R(t))$ is derived from Gibbs-Thomson relation. $H(\cdot)$ is smooth on $(0,\infty)$ satisfying $H(x)=0$ if $x\leq \gamma$, $H(x)=1$ if $x\geq 2\gamma$ and $0\leq H'(x)\leq 2/\gamma$ for all $x\geq 0$. In the case where $\alpha$ 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 $\alpha$ depends on time, we show that $R(t)$ will be also bounded if $\alpha(t)$ is bounded and some sufficient conditions for the disappearance of tumors are given.