Abstract
In this paper we deal with a free boundary problem modeling the growth of nonnecrotic tumors. The tumor is treated as an incompressible fluid, the tissue elasticity is neglected and no chemical inhibitor species are present. We re-express the mathematical model as an operator equation and by using a bifurcation argument we prove that there exist smooth stationary solutions of the problem which are not radially symmetric.
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References
Borisovich A., Friedman A.: Symmetric-breaking bifurcation for free boundary problems. Indiana Univ. Math. J. 54, 927–947 (2005)
Buffoni B., Toland J.: Analytic Theory of Global Bifurcation: An Introduction. Princeton, New Jersey (2003)
Byrne H.M., Chaplain M.A.: Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math. Biosci. 130, 151–181 (1995)
Crandall M.G., Rabinowitz P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)
Cristini V., Lowengrub J., Nie Q.: simulation of tumor growth. J. Math. Biol. 46, 191–224 (2003)
Cui S.B.: Analysis of a free boundary problem modeling tumor growth. Acta Math. Sinica, English Series 21, 1071–1082 (2005)
Cui S.B., Escher J.: Bifurcation analysis of an elliptic free boundary problem modelling the growth of avascular tumors. SIAM J. Math. Anal. 39, 210–235 (2007)
Cui S.B., Escher J.: Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth. Comm. Part. Diff. Eq. 33, 636–655 (2008)
Cui S.B., Escher J., Zhou F.: Bifurcation for a free boundary problem with surface tension modelling the growth of multi-layer tumors. J. Math. Anal. Appl. 337, 443–457 (2008)
De Angelis E., Preziosi L.: Advection diffusion models for solid tumors in vivo and related free-boundary problems. Math. Mod. Meth. Appl. Sci. 10, 379–408 (2000)
Escher J., Matioc A.-V.: Radially symmetric growth of nonnecrotic tumors. NoDEA Nonlinear Differential Equations Appl. 17, 1–20 (2010)
Escher J., Matioc A.-V.: Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors. Discrete Contin. Dyn. Syst. Ser. B 15, 573–596 (2011)
Friedman A., Reitich F.: Analysis of a mathematical model for the growth of tumors. J. Math. Biol. 38, 262–284 (1999)
Friedman A., Reitich F.: Symmetry-breaking bifurcation of analytic solutions to free boundary problems. Trans. Amer. Math. Soc. 353, 1587–1634 (2001)
Greenspan F.P.: On the growth and stability of cell cultures and solid tumors. J. Theor. Biol. 56, 229–242 (1976)
A.-V. Matioc, Modelling and analysis of nonnecrotictumors, Südwestdeutscher Verlag für Hochschulschriften, Saarbrücken, 2009.
Zhou F., Cui S.B.: Bifurcations for a multidimensional free boundary problem modeling the growth of tumor cord. Nonlinear Analysis: real Word Applications 10, 2990–3001 (2009)
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Escher, J., Matioc, AV. Bifurcation analysis for a free boundary problem modeling tumor growth. Arch. Math. 97, 79–90 (2011). https://doi.org/10.1007/s00013-011-0276-8
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DOI: https://doi.org/10.1007/s00013-011-0276-8