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

Published electronically:
February 27, 2008

MathSciNet review:
2415075

Full-text PDF Free Access

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.**Borys V. Bazaliy and Avner Friedman,*A free boundary problem for an elliptic-parabolic system: application to a model of tumor growth*, Comm. Partial Differential Equations**28**(2003), no. 3-4, 517–560. MR**1976462**, 10.1081/PDE-120020486**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.**Helen M. Byrne,*A weakly nonlinear analysis of a model of avascular solid tumour growth*, J. Math. Biol.**39**(1999), no. 1, 59–89. MR**1705626**, 10.1007/s002850050163**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.**Xinfu Chen and Avner Friedman,*A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth*, SIAM J. Math. Anal.**35**(2003), no. 4, 974–986 (electronic). MR**2049029**, 10.1137/S0036141002418388**9.**Michael G. Crandall and Paul H. Rabinowitz,*Bifurcation from simple eigenvalues*, J. Functional Analysis**8**(1971), 321–340. MR**0288640****10.**Marco A. Fontelos and Avner Friedman,*Symmetry-breaking bifurcations of free boundary problems in three dimensions*, Asymptot. Anal.**35**(2003), no. 3-4, 187–206. MR**2011787****11.**Marco A. Fontelos and Avner Friedman,*Symmetry-breaking bifurcations of charged drops*, Arch. Ration. Mech. Anal.**172**(2004), no. 2, 267–294. MR**2058166**, 10.1007/s00205-003-0298-x**12.**Avner Friedman and Bei Hu,*Bifurcation from stability to instability for a free boundary problem arising in a tumor model*, Arch. Ration. Mech. Anal.**180**(2006), no. 2, 293–330. MR**2210911**, 10.1007/s00205-005-0408-z**13.**Avner Friedman and Bei Hu,*Asymptotic stability for a free boundary problem arising in a tumor model*, J. Differential Equations**227**(2006), no. 2, 598–639. MR**2237681**, 10.1016/j.jde.2005.09.008**14.**Avner Friedman and Fernando Reitich,*Analysis of a mathematical model for the growth of tumors*, J. Math. Biol.**38**(1999), no. 3, 262–284. MR**1684873**, 10.1007/s002850050149**15.**Avner Friedman and Fernando Reitich,*Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth*, Trans. Amer. Math. Soc.**353**(2001), no. 4, 1587–1634 (electronic). MR**1806728**, 10.1090/S0002-9947-00-02715-X**16.**Avner Friedman and Fernando Reitich,*Nonlinear stability of a quasi-static Stefan problem with surface tension: a continuation approach*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**30**(2001), no. 2, 341–403. MR**1895715****17.**A. Galindo and P. Pascual,*Quantum mechanics. I*, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1990. Translated from the Spanish by J. D. García and L. Alvarez-Gaumé. MR**1079543****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. Theoret. Biol.**56**(1976), no. 1, 229–242. MR**0429164**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
35R35,
35K55,
35Q80,
35C20,
92C37

Retrieve articles in all journals with MSC (2000): 35R35, 35K55, 35Q80, 35C20, 92C37

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.