  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 {\em Bull. Math. Biol.} (2001). By using a functional approach
and the $C_0$ semigroup theory, we prove that the unique stationary solution of
this model ensured by the work of Cui and Friedman ({\em 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 ({\em Trans. Amer. Math. Soc.}, 2005), and a
similarity transformation.
