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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model
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by Avner Friedman and Bei Hu PDF
Trans. Amer. Math. Soc. 360 (2008), 5291-5342 Request permission


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 $\varepsilon$, $|\varepsilon |$ 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:
  • Bei Hu
  • Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
  • Email:
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 5291-5342
  • MSC (2000): Primary 35R35, 35K55; Secondary 35Q80, 35C20, 92C37
  • DOI:
  • MathSciNet review: 2415075