Analysis of a free boundary problem for tumor growth with angiogenesis and time delays in proliferation. (February 2020) - Xu, Shihe
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 − σ ̄ ) = 0 holds on the boundary, where σ ̄ 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) lim t → ∞ R ( t ) = 0 if lim t → ∞ 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 ) .