## Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth

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- by Avner Friedman and Fernando Reitich PDF
- Trans. Amer. Math. Soc.
**353**(2001), 1587-1634 Request permission

## Abstract:

In this paper we develop a general technique for establishing analyticity of solutions of partial differential equations which depend on a parameter $\varepsilon$. 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 $\varepsilon$.## References

- 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., Birkhรคuser, Boston (1996), 15โ87. - Robert A. Adams,
*Sobolev spaces*, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR**0450957** - Oscar P. Bruno and Peter Laurence,
*Existence of three-dimensional toroidal MHD equilibria with nonconstant pressure*, Comm. Pure Appl. Math.**49**(1996), no.ย 7, 717โ764. MR**1387191**, DOI 10.1002/(SICI)1097-0312(199607)49:7<717::AID-CPA3>3.3.CO;2-6 - Oscar P. Bruno and Fernando Reitich,
*Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain*, Proc. Roy. Soc. Edinburgh Sect. A**122**(1992), no.ย 3-4, 317โ340. MR**1200203**, DOI 10.1017/S0308210500021132 - H.M. Byrne,
*The importance of intercellular adhesion in the development of carcinomas*, IMA J. Math. Appl. Med. and Biol., 14 (1997), 305โ323. - H. M. Byrne,
*The effect of time delays on the dynamics of avascular tumor growth*, Math. Biosci.**144**(1997), no.ย 2, 83โ117. MR**1478080**, DOI 10.1016/S0025-5564(97)00023-0 - H.M. Byrne and M.A.J. Chaplain,
*Growth of necrotic tumors in the presence and absence of inhibitors*, Math. Biosciences, 135 (1996), 187โ216. - Avner Friedman,
*On the regularity of the solutions of nonlinear elliptic and parabolic systems of partial differential equations*, J. Math. Mech.**7**(1958), 43โ59. MR**0118970**, DOI 10.1512/iumj.1958.7.57004 - A. Friedman,
*On the regularity of solutions of nonlinear elliptic and parabolic systems of partial differential equations*, J. Math. Mech., 7 (1958), 43โ60. - A. Friedman and F. Reitich,
*Analysis of a mathematical model for the growth of tumors*, J. Math. Biol., 38 (1999), 262โ284. - A. Friedman and F. Reitich,
*Nonlinear stability of a quasi-static Stefan problem with surface tension: a continuation approach*, to appear. - H. P. Greenspan,
*On the growth and stability of cell cultures and solid tumors*, J. Theoret. Biol.**56**(1976), no.ย 1, 229โ242. MR**429164**, DOI 10.1016/S0022-5193(76)80054-9 - 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), 147โ157. - David H. Sattinger,
*Group-theoretic methods in bifurcation theory*, Lecture Notes in Mathematics, vol. 762, Springer, Berlin, 1979. With an appendix entitled โHow to find the symmetry group of a differential equationโ by Peter Olver. MR**551626** - Joel Smoller,
*Shock waves and reaction-diffusion equations*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR**688146** - Albert Eagle,
*Series for all the roots of the equation $(z-a)^m=k(z-b)^n$*, Amer. Math. Monthly**46**(1939), 425โ428. MR**6**, DOI 10.2307/2303037

## 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
- Received by editor(s): August 17, 1999
- Published electronically: November 21, 2000
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1806728