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In this paper, we study a free boundary problem modeling solid tumor growthwith vasculature which supplies nutrients to the tumor; this is characterizedin the Robin boundary condition. It was recently established [Discrete Cont.Dyn. Syst. 39 (2019) 2473-2510] that for this model, there exists a thresholdvalue $\mu^\ast$ such that the unique radially symmetric stationary solution islinearly stable under non-radial perturbations for $0<\mu<\mu^\ast$ andlinearly unstable for $\mu>\mu^\ast$. In this paper we further study thenonlinear stability of the radially symmetric stationary solution, whichintroduces a significant mathematical difficulty: the center of the limitingsphere is not known in advance owing to the perturbation of mode 1 terms. Weprove a new fixed point theorem to solve this problem, and finally obtain thatthe radially symmetric stationary solution is nonlinearly stable for$0<\mu<\mu^\ast$ when neglecting translations.

Asymptotic stability for a free boundary tumor model with angiogenesis