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

 

Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model


Authors: Avner Friedman and Bei Hu
Journal: Trans. Amer. Math. Soc. 360 (2008), 5291-5342
MSC (2000): Primary 35R35, 35K55; Secondary 35Q80, 35C20, 92C37
Published electronically: February 27, 2008
MathSciNet review: 2415075
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Abstract: We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth. For any positive number $ R$ there exists a radially symmetric stationary solution with free boundary $ r=R$. The system depends on a positive parameter $ \mu$, and for a sequence of values $ \mu_2<\mu_3<\cdots$ there also exist branches of symmetric-breaking stationary solutions, parameterized by $ \varepsilon$, $ \vert\varepsilon\vert$ small, which bifurcate from these values. In particular, for $ \mu=\mu(\varepsilon)$ near $ \mu_2$ the free boundary has the form $ r=R+\varepsilon Y_{2,0}(\theta)+O(\varepsilon^2)$ where $ Y_{2,0}$ is the spherical harmonic of mode $ (2,0)$. It was recently proved by the authors that the stationary solution is asymptotically stable for any $ 0<\mu<\mu_*$, but linearly unstable if $ \mu>\mu_*$, where $ \mu_*=\mu_2$ if $ R>\bar R$ and $ \mu_*<\mu_2$ if $ R<\bar R$; $ \bar R\approx 0.62207$. In this paper we prove that for $ R>\bar R$ each of the stationary solutions which bifurcates from $ \mu=\mu_2$ is linearly stable if $ \varepsilon>0$ and linearly unstable if $ \varepsilon<0$. We also prove, for $ R<\bar R$, that the point $ \mu=\mu_*$ is a Hopf bifurcation, in the sense that the linearized time-dependent problem has a family of solutions which are asymptotically periodic in $ t$.


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Additional Information

Avner Friedman
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
Email: afriedman@mbi.osu.edu

Bei Hu
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: b1hu@nd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04468-1
PII: S 0002-9947(08)04468-1
Keywords: Free boundary problems, stationary solution, stability, instability, bifurcation, symmetry-breaking, tumor cell
Received by editor(s): February 24, 2005
Received by editor(s) in revised form: September 26, 2005, and September 6, 2006
Published electronically: February 27, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.