Symmetrybreaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth
Authors:
Avner Friedman and Fernando Reitich
Journal:
Trans. Amer. Math. Soc. 353 (2001), 15871634
MSC (1991):
Primary 35B32, 35R35; Secondary 35B30, 35B60, 35C10, 35J85, 35Q80, 92C15, 95C15
Published electronically:
November 21, 2000
MathSciNet review:
1806728
Fulltext PDF Free Access
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Abstract: In this paper we develop a general technique for establishing analyticity of solutions of partial differential equations which depend on a parameter . The technique is worked out primarily for a free boundary problem describing a model of a stationary tumor. We prove the existence of infinitely many branches of symmetrybreaking solutions which bifurcate from any given radially symmetric steady state; these asymmetric solutions are analytic jointly in the spatial variables and in .
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 A. Friedman, On the regularity of solutions of nonlinear elliptic and parabolic systems of partial differential equations, J. Math. Mech., 7 (1958), 4360.
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 A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262284. CMP 99:11
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 A. Friedman and F. Reitich, Nonlinear stability of a quasistatic Stefan problem with surface tension: a continuation approach, to appear.
 12.
 H.P. Greenspan, On the growth and stability of cell cultures and solid tumors, Theor. Biol., 56 (1976), 229242. MR 55:2183
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 D.L.S. McElwain and L.E. Morris, Apoptosis as a volume loss mechanism in mathematical models of solid tumor growth, Math. Biosciences, 39 (1978), 147157.
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 D.H Sattinger, GroupTheoretic Methods in Bifurcation Theory, Lecture Notes in Mathematics, 762, SpringerVerlag, Berlin (1979). MR 81e:58022
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 J. Smoller, Shock Waves and ReactionDiffusion Equations, SpringerVerlag, New York (1983). MR 84d:35002
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 G.N. Watson, A Treatise on the Theory of Bessel Functions, Second Edition, Cambridge University Press (1944). MR 6:64a
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Additional Information
Avner Friedman
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email:
friedman@math.umn.edu
Fernando Reitich
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email:
reitich@math.umn.edu
DOI:
http://dx.doi.org/10.1090/S000299470002715X
PII:
S 00029947(00)02715X
Keywords:
Free boundary problem,
steady states,
bifurcation,
symmetrybreaking,
analytic solutions,
tumor growth
Received by editor(s):
August 17, 1999
Published electronically:
November 21, 2000
Article copyright:
© Copyright 2000
American Mathematical Society
