Asymptotic stability for a free boundary tumor model with angiogenesis

In this paper, we study a free boundary problem modeling solid tumor growth with vasculature which supplies nutrients to the tumor; this is characterized in the Robin boundary condition. It was recently established [Discrete Cont. Dyn. Syst. 39 (2019) 2473-2510] that for this model, there exists a threshold value $μ^\ast$ such that the unique radially symmetric stationary solution is linearly stable under non-radial perturbations for $0<μ<μ^\ast$ and linearly unstable for $μ>μ^\ast$. In this paper we further study the nonlinear stability of the radially symmetric stationary solution, which introduces a significant mathematical difficulty: the center of the limiting sphere is not known in advance owing to the perturbation of mode 1 terms. We prove a new fixed point theorem to solve this problem, and finally obtain that the radially symmetric stationary solution is nonlinearly stable for $0<μ<μ^\ast$ when neglecting translations.

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