American Institute of Mathematical Sciences

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June  2009, 24(2): 625-651. doi: 10.3934/dcds.2009.24.625

Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations

Junde Wu 1, and  Shangbin Cui 2,

1. 

Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China
2. 

Institute of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275

Received  July 2008 Revised  November 2008 Published  March 2009

We study a free boundary problem modelling the growth of non-necrotic tumors with fluid-like tissues. The fluid velocity satisfies Stokes equations with a source determined by the proliferation rate of tumor cells which depends on the concentration of nutrients, subject to a boundary condition with stress tensor effected by surface tension. It is easy to prove that this problem has a unique radially symmetric stationary solution. By using a functional approach, we prove that there exists a threshold value γ* > 0 for the surface tension coefficient $\gamma$, such that in the case γ > γ* this radially symmetric stationary solution is asymptotically stable under small non-radial perturbations, whereas in the opposite case it is unstable.
Keywords: stationary solution, Free boundary problem, asymptotic stability., Stokes equations, tumor growth.
Mathematics Subject Classification: Primary: 35R35, 35B35; Secondary: 76D2.
Citation: Junde Wu, Shangbin Cui. Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 625-651. doi: 10.3934/dcds.2009.24.625
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