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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with fluid-like tissue under the action of inhibitors


Authors: Junde Wu and Fujun Zhou
Journal: Trans. Amer. Math. Soc. 365 (2013), 4181-4207
MSC (2010): Primary 35B40, 35R35; Secondary 35Q92, 92C37
Published electronically: January 28, 2013
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Abstract: In this paper we study a free boundary problem modeling the growth of tumors with fluid-like tissue under the action of inhibitors. The model includes two elliptic equations describing the concentration of nutrients and inhibitors, respectively, and a Stokes equation for the fluid velocity and internal pressure. By employing the functional approach, analytic semigroup theory and Cui's local phase theorem for parabolic differential equations with invariance, we prove that if a radial stationary solution is asymptotically stable under radial perturbations, then there exists a non-negative threshold value $ \gamma _*$ such that if $ \gamma >\gamma _*$, then it keeps asymptotically stable under non-radial perturbations. While if $ 0<\gamma <\gamma _*$, then the radial stationary solution is unstable and, in particular, there exists a center-stable manifold such that if the transient solution exists globally and is contained in a sufficiently small neighborhood of the radial stationary solution, then it converges exponentially to this radial stationary solution (modulo translations) and its translation lies on the center-stable manifold. The result indicates an interesting phenomenon that an increasing inhibitor uptake has a positive effect on the tumor's treatment and can promote the tumor's stability.


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Additional Information

Junde Wu
Affiliation: Department of Mathematics, Soochow University, Suzhou, Jiangsu 215006, People’s Republic of China
Email: wjdmath@yahoo.com.cn

Fujun Zhou
Affiliation: Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, People’s Republic of China
Email: zhoufujunht@yahoo.com.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05779-0
PII: S 0002-9947(2013)05779-0
Keywords: Free boundary problem, tumor growth, Stokes equation, radial stationary solution, asymptotic stability
Received by editor(s): May 23, 2011
Received by editor(s) in revised form: November 18, 2011
Published electronically: January 28, 2013
Additional Notes: This work was supported by the National Natural Science Foundation of China under the grant numbers 10901057 and 11001192, the Doctoral Foundation of Education Ministry of China under the grant numbers 200805611027 and 20103201120017, the Natural Science Fund for Colleges and Universities in Jiangsu Province under the grant number 10KJB110008, and the Fundamental Research Funds for the Central Universities of SCUT under the grant number 2012ZZ0072.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.