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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

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

Pages: 625 - 651, Volume 24, Issue 2, June 2009

doi:10.3934/dcds.2009.24.625       Abstract        Full Text (362.9K)       Related Articles

Junde Wu - Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China (email)
Shangbin Cui - Institute of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China (email)

Abstract: 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:  Free boundary problem, tumor growth, Stokes equations, stationary solution, asymptotic stability.
Mathematics Subject Classification:  Primary: 35R35, 35B35; Secondary: 76D27.

Received: July 2008;      Revised: November 2008;      Published: March 2009.