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In this paper we study a free boundary problem for the growth of multi-layertumors in necrotic phase. The tumor region is strip-like and divided intonecrotic region and proliferating region with two free boundaries. The upperfree boundary is tumor surface and governed by a Stefan condition. The lowerfree boundary is the interface separating necrotic region from proliferatingregion, its evolution is implicit and intrinsically governed by an obstacleproblem. We prove that the problem has a unique flat stationary solution, andthere exists a positive constant $\gamma_*$, such that the flat stationarysolution is asymptotically stable for cell-to-cell adhesiveness$\gamma>\gamma_*$, and unstable for $0<\gamma<\gamma_*$.

Asymptotic behavior of a free boundary problem for the growth of multi-layer tumors in necrotic phase