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December  2011, 15(3): 573-596. doi: 10.3934/dcdsb.2011.15.573

Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors

Joachim Escher 1, and  Anca-Voichita Matioc 1,

1. 

Institute for Applied Mathematics, Leibniz University Hanover, Welfengarten 1, 30167 Hanover, Germany, Germany

Received  February 2010 Revised  April 2010 Published  February 2011

We study a moving boundary problem describing the growth of nonnecrotic tumors in different regimes of vascularisation. This model consists of two decoupled Dirichlet problem, one for the rate at which nutrient is added to the tumor domain and one for the pressure inside the tumor. These variables are coupled by a relation which describes the dynamic of the boundary. By re-expressing the problem as an abstract evolution equation, we prove local well-posedness in the small Hölder spaces context. Further on, we use the principle of linearised stability to characterise the stability properties of the unique radially symmetric equilibrium of the problem.
Keywords: Tumor growth, Stability., Moving boundary problem, Well-posedness.
Mathematics Subject Classification: Primary: 35B35; Secondary: 35B40, 35K5.
Citation: Joachim Escher, Anca-Voichita Matioc. Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 573-596. doi: 10.3934/dcdsb.2011.15.573
References:
[1]

H. Amann, "Linear and Quasilinear Parabolic Problems," Volume I Birkhäuser, Basel, 1995.  Google Scholar

[2]

W. Arendt and S. Bu, Operator-valued Fourier multipliers on periodic Besov spaces and applications, Proceedings of the Edinburgh Mathematical Society, 47 (2004), 15-33. doi: 10.1017/S0013091502000378.  Google Scholar

[3]

N. Bellomo, N. K. Li and P. K. Maini, On the foundations of cancer modelling: Selected topics, speculations, and perspectives, Mathematical Models and Methods in Applied Sciences, 18 (2008), 593-647. doi: 10.1142/S0218202508002796.  Google Scholar

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A. Borisovich and A. Friedman, Symmetric-breaking bifurcation for free boundary problems, Indiana Univ. Math. J., 54 (2005), 927-947. doi: 10.1512/iumj.2005.54.2473.  Google Scholar

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H. M. Byrne and M. A. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181. doi: 10.1016/0025-5564(94)00117-3.  Google Scholar

[6]

V. Cristini, J. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth, Journal of Mathematical Biology, 46 (2003), 191-224. doi: 10.1007/s00285-002-0174-6.  Google Scholar

[7]

S. B. Cui, Analysis of a free boundary problem modeling tumor growth, Acta Mathematica Sinica, English Series, 21 (2005), 1071-1082. doi: 10.1007/s10114-004-0483-3.  Google Scholar

[8]

S. B. Cui and J. Escher, Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth, Comm. Part. Diff. Eq., 33 (2008), 636-655.  Google Scholar

[9]

S. B. Cui and J. Escher, Bifurcation analysis of an elliptic free boundary problem modelling the growth of avascular tumors, SIAM J. Math. Anal., 39 (2007), 210-235. doi: 10.1137/060657509.  Google Scholar

[10]

S. B. Cui and A. Friedman, A hyperbolic free boundary problem modelling tumor growth, Interface Free Bound, 5 (2003), 159-181. doi: 10.4171/IFB/76.  Google Scholar

[11]

G. Da Prato and P. Grisvard, Equations d'évolution abstraites nonlinéaires de type parabolique, Ann. Mat. Pura Appl., 120 (1979), 329-326. doi: 10.1007/BF02411952.  Google Scholar

[12]

J. Escher and A. V. Matioc, Radially symmetric growth of nonnecrotic tumors, Nonlinear Differential Equations and Applications, 17 (2010), 1-20. doi: 10.1007/s00030-009-0037-6.  Google Scholar

[13]

J. Escher, A. V. Matioc and B. V. Matioc, Classical solutions and stability results for Stokesian Hele-Shaw flows, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 5, 9 (2010), 325-349. Google Scholar

[14]

A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262-284. doi: 10.1007/s002850050149.  Google Scholar

[15]

A. Friedman and F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems, Trans. Amer. Math. Soc., 353 (2001), 1587-1634. doi: 10.1090/S0002-9947-00-02715-X.  Google Scholar

[16]

A. Friedman, Cancer models and their mathematical analysis, Lect. Notes Math., 1872 (2006), 223-246. doi: 10.1007/11561606_6.  Google Scholar

[17]

D. Gilbarg and T. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, New York, 2001.  Google Scholar

[18]

H. P. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theor. Biol., 56 (1976), 229-242. doi: 10.1016/S0022-5193(76)80054-9.  Google Scholar

[19]

E. I. Hanzawa, Classical solutions of the Stefan problem, Tôhoku Math. J., 33 (1981), 297-335.  Google Scholar

[20]

T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, Berlin Heidelberg, 1995.  Google Scholar

[21]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, Basel, 1995.  Google Scholar

[22]

A. V. Matioc, "Modelling and Analysis of Nonnecrotic Tumors," Südwestdeutcher Verlag für Hochschulschriften, Saarbrücken, 2009. Google Scholar

[23]

E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, Journal of Mathematical Analysis and Applications, 107 (1985), 16-66. doi: 10.1016/0022-247X(85)90353-1.  Google Scholar

show all references

References:
[1]

H. Amann, "Linear and Quasilinear Parabolic Problems," Volume I Birkhäuser, Basel, 1995.  Google Scholar

[2]

W. Arendt and S. Bu, Operator-valued Fourier multipliers on periodic Besov spaces and applications, Proceedings of the Edinburgh Mathematical Society, 47 (2004), 15-33. doi: 10.1017/S0013091502000378.  Google Scholar

[3]

N. Bellomo, N. K. Li and P. K. Maini, On the foundations of cancer modelling: Selected topics, speculations, and perspectives, Mathematical Models and Methods in Applied Sciences, 18 (2008), 593-647. doi: 10.1142/S0218202508002796.  Google Scholar

[4]

A. Borisovich and A. Friedman, Symmetric-breaking bifurcation for free boundary problems, Indiana Univ. Math. J., 54 (2005), 927-947. doi: 10.1512/iumj.2005.54.2473.  Google Scholar

[5]

H. M. Byrne and M. A. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181. doi: 10.1016/0025-5564(94)00117-3.  Google Scholar

[6]

V. Cristini, J. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth, Journal of Mathematical Biology, 46 (2003), 191-224. doi: 10.1007/s00285-002-0174-6.  Google Scholar

[7]

S. B. Cui, Analysis of a free boundary problem modeling tumor growth, Acta Mathematica Sinica, English Series, 21 (2005), 1071-1082. doi: 10.1007/s10114-004-0483-3.  Google Scholar

[8]

S. B. Cui and J. Escher, Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth, Comm. Part. Diff. Eq., 33 (2008), 636-655.  Google Scholar

[9]

S. B. Cui and J. Escher, Bifurcation analysis of an elliptic free boundary problem modelling the growth of avascular tumors, SIAM J. Math. Anal., 39 (2007), 210-235. doi: 10.1137/060657509.  Google Scholar

[10]

S. B. Cui and A. Friedman, A hyperbolic free boundary problem modelling tumor growth, Interface Free Bound, 5 (2003), 159-181. doi: 10.4171/IFB/76.  Google Scholar

[11]

G. Da Prato and P. Grisvard, Equations d'évolution abstraites nonlinéaires de type parabolique, Ann. Mat. Pura Appl., 120 (1979), 329-326. doi: 10.1007/BF02411952.  Google Scholar

[12]

J. Escher and A. V. Matioc, Radially symmetric growth of nonnecrotic tumors, Nonlinear Differential Equations and Applications, 17 (2010), 1-20. doi: 10.1007/s00030-009-0037-6.  Google Scholar

[13]

J. Escher, A. V. Matioc and B. V. Matioc, Classical solutions and stability results for Stokesian Hele-Shaw flows, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 5, 9 (2010), 325-349. Google Scholar

[14]

A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262-284. doi: 10.1007/s002850050149.  Google Scholar

[15]

A. Friedman and F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems, Trans. Amer. Math. Soc., 353 (2001), 1587-1634. doi: 10.1090/S0002-9947-00-02715-X.  Google Scholar

[16]

A. Friedman, Cancer models and their mathematical analysis, Lect. Notes Math., 1872 (2006), 223-246. doi: 10.1007/11561606_6.  Google Scholar

[17]

D. Gilbarg and T. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, New York, 2001.  Google Scholar

[18]

H. P. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theor. Biol., 56 (1976), 229-242. doi: 10.1016/S0022-5193(76)80054-9.  Google Scholar

[19]

E. I. Hanzawa, Classical solutions of the Stefan problem, Tôhoku Math. J., 33 (1981), 297-335.  Google Scholar

[20]

T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, Berlin Heidelberg, 1995.  Google Scholar

[21]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, Basel, 1995.  Google Scholar

[22]

A. V. Matioc, "Modelling and Analysis of Nonnecrotic Tumors," Südwestdeutcher Verlag für Hochschulschriften, Saarbrücken, 2009. Google Scholar

[23]

E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, Journal of Mathematical Analysis and Applications, 107 (1985), 16-66. doi: 10.1016/0022-247X(85)90353-1.  Google Scholar

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