Abstract
This paper is concerned with a free boundary problem modeling the growth of a spherically symmetric tumor with angiogenesis. The unknown nutrient concentration \(\sigma =\sigma (r,t)\) occupies the unknown tumor region \(r< R(t)\) and satisfies a nonlinear reaction diffusion equation, and the unknown tumor radius \(R=R(t)\) satisfies a nonlinear integro-differential equation. Unlike existing literatures on this topic where Dirichlet boundary condition for \(\sigma \) is imposed, in this paper the model uses the Robin boundary condition for \(\sigma \). We prove existence and uniqueness of a global in-time classical solution (\(\sigma (r,t),R(t)\)) for arbitrary \(c>0\) and establish asymptotic stability of the unique stationary solution (\(\sigma _{s}(r),R_{s}\)) for sufficiently small \(c\), where \(c\) is a positive constant reflecting the ratio between nutrient diffusion scale and the tumor cell-doubling scale.
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The authors are very happy to acknowledge their sincere thanks to the anonymous referee for valuable suggestions on modification of this manuscript.
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This work is supported by National Natural Science Foundation of China under the grant number 11571381.
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Zhuang, Y., Cui, S. Analysis of a Free Boundary Problem Modeling the Growth of Spherically Symmetric Tumors with Angiogenesis. Acta Appl Math 161, 153–169 (2019). https://doi.org/10.1007/s10440-018-0208-8
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DOI: https://doi.org/10.1007/s10440-018-0208-8