NDLI: Asymptotic Stability of Stationary Solutions of a Free Boundary Problem Modeling the Growth of Tumors with Fluid Tissues

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Wu, Junde Cui, Shangbin
Copyright Year	2009
Abstract	This paper aims at proving asymptotic stability of the radial stationary solution of a free boundary problem modeling the growth of nonnecrotic tumors with fluid-like tissues. In a previous paper we considered the case where the nutrient concentration $\sigma$ satisfies the stationary diffusion equation $\Delta\sigma=f(\sigma)$ and proved that there exists a threshold value $\gamma_>0$ for the surface tension coefficient $\gamma$, such that the radial stationary solution is asymptotically stable in case $\gamma>\gamma_$, while unstable in case $\gamma<\gamma_$. In this paper we extend this result to the case where $\sigma$ satisfies the nonstationary diffusion equation $\varepsilon\partial_t\sigma=\Delta\sigma-f(\sigma)$. We prove that for the same threshold value $\gamma_$ as above, for every $\gamma>\gamma_$, there is a corresponding constant $\varepsilon_0(\gamma)>0$ such that for any $0<\varepsilon<\varepsilon_0(\gamma)$, the radial stationary solution is asymptotically stable with respect to small enough nonradial perturbations, while for $0<\gamma<\gamma_$ and $\varepsilon$ sufficiently small, it is unstable under nonradial perturbations.
Starting Page	391
Ending Page	414
Page Count	24
File Format	PDF
ISSN	00361410
DOI	10.1137/080726550
e-ISSN	10957154
Journal	SIAM Journal on Mathematical Analysis (SJMAAH)
Issue Number	1
Volume Number	41
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-05-13
Access Restriction	Subscribed
Subject Keyword	Stability free boundary problem tumor growth radial stationary solution Free boundary problems Other free-boundary flows; Hele-Shaw flows asymptotic stability Stokes equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Mathematics

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