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

DOI:
https://doi.org/10.1090/S0002-9947-2013-05779-0

Published electronically:
January 28, 2013

MathSciNet review:
3055693

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Abstract | References | Similar Articles | Additional Information

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 such that if , then it keeps asymptotically stable under non-radial perturbations. While if , 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:
https://doi.org/10.1090/S0002-9947-2013-05779-0

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.