Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model

Authors:
Avner Friedman and Bei Hu

Journal:
Trans. Amer. Math. Soc. **360** (2008), 5291-5342

MSC (2000):
Primary 35R35, 35K55; Secondary 35Q80, 35C20, 92C37

DOI:
https://doi.org/10.1090/S0002-9947-08-04468-1

Published electronically:
February 27, 2008

MathSciNet review:
2415075

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Abstract | References | Similar Articles | Additional Information

Abstract: 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 there exists a radially symmetric stationary solution with free boundary . The system depends on a positive parameter , and for a sequence of values there also exist branches of symmetric-breaking stationary solutions, parameterized by , small, which bifurcate from these values. In particular, for near the free boundary has the form where is the spherical harmonic of mode . It was recently proved by the authors that the stationary solution is asymptotically stable for any , but linearly unstable if , where if and if ; . In this paper we prove that for each of the stationary solutions which bifurcates from is linearly stable if and linearly unstable if . We also prove, for , that the point is a Hopf bifurcation, in the sense that the linearized time-dependent problem has a family of solutions which are asymptotically periodic in .

**1.**B. BAZALLY AND A. FRIEDMAN,*A free boundary problem for an elliptic-parabolic system: Application to a model of tumor growth*, Comm. Partial Diff. Eq., Vol. 28, 2003, pp. 517-560. MR**1976462 (2004c:35420)****2.**H.M. BYRNE,*The importance of intercellular adhesion in the development of carcinomas*, IMA J. Math. Appl. Med. Biol., Vol. 14, 1997, pp. 305-323.**3.**H.M. BYRNE,*A weakly nonlinear analysis of a model of avascular solid tumour growth*, J. Math. Biol., Vol. 39, 1999, pp. 59-89. MR**1705626 (2000i:92011)****4.**H.M. BYRNE AND M.A.J. CHAPLAIN,*Modelling the role of cell-cell adhesion in the growth and development of carcinomas*, Mathl. Comput. Modelling, Vol. 12, 1996, pp. 1-17.**5.**H.M. BYRNE AND M.A.J. CHAPLAIN,*Growth of nonnecrotic tumors in the presence and absence of inhibitors*, Math. Biosci., Vol. 130, 1995, pp. 151-181.**6.**H.M. BYRNE AND M.A.J. CHAPLAIN,*Growth of nonnecrotic tumors in the presence and absence of inhibitors*, Math. Biosci., Vol. 135, 1996, pp. 187-216.**7.**M.A.J. CHAPLAIN,*The development of a spatial pattern in a model for cancer growth*, Experimental and Theoretical Advances in Biological Pattern Formation (H.G. Othmer, P.K. Maini, and J.D. Murray, eds), Plenum Press, 1993, pp. 45-60.**8.**X. CHEN AND A. FRIEDMAN,*A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth*, SIAM J. Math. Anal., Vol. 35, 2003, pp. 974-986. MR**2049029 (2005f:35333)****9.**M.G. CRANDALL AND P.H. RABINOWITZ,*Bifurcation from simple eigenvalies*, J. Functional Analysis, Vol. 8, 1971, pp. 321-340. MR**0288640 (44:5836)****10.**M. FONTELOS AND A. FRIEDMAN,*Symmetry-breaking bifurcations of free boundary problems in three dimensions*, Asymptotic Analysis, Vol. 35, 2003, pp. 187-206. MR**2011787 (2005a:35290)****11.**M. FONTELOS AND A. FRIEDMAN,*Symmetry-breaking bifurcations of charged drops*, Arch. Rat. Mech. Anal., Vol. 172, 2004, pp. 267-294. MR**2058166 (2005b:76140)****12.**A. FRIEDMAN AND BEI HU,*Bifurcation from stability to instability for a free boundary problem arising in a tumor model*, Arch. Rat. Mech. Anal., Vol. 180, No. 2, 2006, pp. 293-330. MR**2210911 (2006j:35246)****13.**A. FRIEDMAN AND BEI HU,*Asymptotic stability for a free boundary problem arising in a tumor model*, J. Diff. Eqn., Vol. 227, No. 2, 2006, pp. 598-639. MR**2237681****14.**A. FRIEDMAN AND F. REITICH,*Analysis of a mathematical model for the growth of tumors*, J. Math. Biology, Vol. 38, 1999, pp. 262-284. MR**1684873 (2001f:92011)****15.**A. FRIEDMAN AND F. REITICH,*Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth*, Trans. Amer. Math. Soc., Vol. 353, 2001, pp. 1587-1634. MR**1806728 (2002a:35208)****16.**A. FRIEDMAN AND F. REITICH,*Nonlinear stability of a quasi-static Stefan problem with surface tension: A continuation approach*, Ann. Scoula Norm. Sup. Pisa Cl. Sci., Vol. 30(4), 2001, pp. 341-403. MR**1895715 (2003e:35326)****17.**A. GALINDO AND P. PASCUAL,*Quantum Mechanics, Vol. I*, Springer, New York, 1990. MR**1079543 (91k:81001)****18.**H.P. GREENSPAN,*Models for the growth of a solid tumor by diffusion*, Studies Appl. Math, Vol. 52, 1972, pp. 317-340.**19.**H.P. GREENSPAN,*On the growth and stability of cell cultures and solid tumors*, J. Theoretical Biology, Vol. 56, 1976, pp. 229-242. MR**0429164 (55:2183)**

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Additional Information

**Avner Friedman**

Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210

Email:
afriedman@mbi.osu.edu

**Bei Hu**

Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Email:
b1hu@nd.edu

DOI:
https://doi.org/10.1090/S0002-9947-08-04468-1

Keywords:
Free boundary problems,
stationary solution,
stability,
instability,
bifurcation,
symmetry-breaking,
tumor cell

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

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.