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

Friedman, Avner; Reitich, Fernando

doi:10.1090/S0002-9947-00-02715-X

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., Birkhäuser, Boston (1996), 15–87.
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. 7, 717–764. 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. 3-4, 317–340. 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), 305–323.
H. M. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci. 144 (1997), no. 2, 83–117. 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), 187–216.
Avner Friedman, On the regularity of the solutions of nonlinear elliptic and parabolic systems of partial differential equations, J. Math. Mech. 7 (1958), 43–59. 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), 43–60.
A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262–284.
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. 1, 229–242. 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), 147–157.
David H. Sattinger, Group-theoretic methods in bifurcation theory, Lecture Notes in Mathematics, vol. 762, Springer, Berlin, 1979. With an appendix entitled “How to find the symmetry group of a differential equation” 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), 425–428. MR 6, DOI 10.2307/2303037