Symmetry-breaking 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), 1587-1634

MSC (1991):
Primary 35B32, 35R35; Secondary 35B30, 35B60, 35C10, 35J85, 35Q80, 92C15, 95C15

DOI:
https://doi.org/10.1090/S0002-9947-00-02715-X

Published electronically:
November 21, 2000

MathSciNet review:
1806728

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 symmetry-breaking solutions which bifurcate from any given radially symmetric steady state; these asymmetric solutions are analytic jointly in the spatial variables and in .

**1.**J.A. Adam, General aspects of modeling tumor growth and immune response, in*A Survey of Models for Tumor-Immune System Dynamics*, J.A. Adam and N. Bellomo, eds., Birkhäuser, Boston (1996), 15-87.**2.**R.A. Adams,*Sobolev Spaces*, Academic Press, New York (1975). MR**56:9247****3.**O. Bruno and P. Laurence,*Existence of three-dimensional toroidal MHD equilibria with nonconstant pressure*, Comm. Pure Appl. Math., 49 (1996), 717-764. MR**97g:35137****4.**O. Bruno and F. Reitich,*Solution of the boundary value problem for the Helmholz equation via variation of the boundary into the complex domain*, Proc. Roy. Soc. Edinburgh, Sec. A 122 (1992), 317-340. MR**94e:35131****5.**H.M. Byrne,*The importance of intercellular adhesion in the development of carcinomas*, IMA J. Math. Appl. Med. and Biol., 14 (1997), 305-323.**6.**H.M. Byrne,*The effect of time delays on the dynamics of avascular tumor growth*, Math. Biosciences, 144 (1997), 83-117. MR**98h:92014****7.**H.M. Byrne and M.A.J. Chaplain,*Growth of necrotic tumors in the presence and absence of inhibitors*, Math. Biosciences, 135 (1996), 187-216.**8.**M.A.J. Chaplain,*The development of a spatial pattern in a model for cancer growth*, in*Experimental and Theoretical Advances in Biological Pattern Formation*, H.G. Othmer, P.K. Maini and J.D. Murray eds., Plenum Press (1993), 45-59. MR**22:9739****9.**A. Friedman,*On the regularity of solutions of nonlinear elliptic and parabolic systems of partial differential equations*, J. Math. Mech., 7 (1958), 43-60.**10.**A. Friedman and F. Reitich,*Analysis of a mathematical model for the growth of tumors*, J. Math. Biol., 38 (1999), 262-284. CMP**99:11****11.**A. Friedman and F. Reitich,*Nonlinear stability of a quasi-static 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), 229-242. MR**55:2183****13.**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), 147-157.**14.**D.H Sattinger,*Group-Theoretic Methods in Bifurcation Theory, Lecture Notes in Mathematics*, 762, Springer-Verlag, Berlin (1979). MR**81e:58022****15.**J. Smoller,*Shock Waves and Reaction-Diffusion Equations*, Springer-Verlag, New York (1983). MR**84d:35002****16.**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:
https://doi.org/10.1090/S0002-9947-00-02715-X

Keywords:
Free boundary problem,
steady states,
bifurcation,
symmetry-breaking,
analytic solutions,
tumor growth

Received by editor(s):
August 17, 1999

Published electronically:
November 21, 2000

Article copyright:
© Copyright 2000
American Mathematical Society