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Stability and bifurcation analysis of a free boundary problem modelling multi-layer tumours with Gibbs–Thomson relation

Published online by Cambridge University Press:  10 April 2015

FUJUN ZHOU  and
JUNDE WU
FUJUN ZHOU
Affiliation:
Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, P.R. China email: fujunht@scut.edu.cn
JUNDE WU
Affiliation:
Department of Mathematics, Soochow University, Suzhou, Jiangsu 215006, P.R. China email: wujund@suda.edu.cn
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Abstract

Of concern is the stability and bifurcation analysis of a free boundary problem modelling the growth of multi-layer tumours. A remarkable feature of this problem lies in that the free boundary is imposed with nonlinear boundary conditions, where a Gibbs–Thomson relation is taken into account. By employing a functional approach, analytic semigroup theory and bifurcation theory, we prove that there exists a positive threshold value γ* of surface tension coefficient γ such that if γ > γ* then the unique flat stationary solution is asymptotically stable under non-flat perturbations, while for γ < γ* this unique flat stationary solution is unstable and there exists a series of non-flat stationary solutions bifurcating from it. The result indicates a significant phenomenon that a smaller value of surface tension coefficient γ may make tumours more aggressive.

Keywords

35R3535B3535B3235Q92free boundary problemstabilitybifurcationtumour model

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Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 4 , August 2015 , pp. 401 - 425
Copyright
Copyright © Cambridge University Press 2015 

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