Asymptotic Stability of the Stationary Solution for a Hyperbolic Free Boundary Problem Modeling Tumor Growth

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In this paper we study asymptotic behavior of solutions for a free boundary problem modeling the growth of tumors containing two species of cells: proliferating cells and quiescent cells. This tumor model was proposed by Pettet et al in Bull. Math. Biol. (2001). By using a functional approach and the C₀ semigroup theory, we prove that the unique stationary solution of this model ensured by the work of Cui and Friedman (Trans. Amer. Math. Soc., 2003) is locally asymptotically stable in certain function spaces. Key techniques used in the proof include an improvement of the linear estimate obtained by the work of Chen et al (Trans. Amer. Math. Soc., 2005), and a similarity transformation.