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Asymptotic stability for a free boundary tumor model with angiogenesis. (English) Zbl 07269195
Summary: 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 [Y. Huang et al., Discrete Contin. Dyn. Syst. 39, No. 5, 2473–2510 (2019; Zbl 1414.35236)] that for this model, there exists a threshold value $$\mu^\ast$$ such that the unique radially symmetric stationary solution is linearly stable under non-radial perturbations for $$0 < \mu < \mu^\ast$$ and linearly unstable for $$\mu > \mu^\ast$$. 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 $$0 < \mu < \mu^\ast$$ when neglecting translations.
##### MSC:
 35Q92 PDEs in connection with biology, chemistry and other natural sciences 92C37 Cell biology 35B35 Stability in context of PDEs 35R35 Free boundary problems for PDEs 35B40 Asymptotic behavior of solutions to PDEs 35C10 Series solutions to PDEs
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