# American Institute of Mathematical Sciences

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

 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.
Citation: Joachim Escher, Anca-Voichita Matioc. Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [16] A. Friedman, Cancer models and their mathematical analysis, Lect. Notes Math., 1872 (2006), 223-246. doi: 10.1007/11561606_6. [17] D. Gilbarg and T. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, New York, 2001. [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. [19] E. I. Hanzawa, Classical solutions of the Stefan problem, Tôhoku Math. J., 33 (1981), 297-335. [20] T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, Berlin Heidelberg, 1995. [21] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, Basel, 1995. [22] A. V. Matioc, "Modelling and Analysis of Nonnecrotic Tumors," Südwestdeutcher Verlag für Hochschulschriften, Saarbrücken, 2009. [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.

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##### References:
 [1] H. Amann, "Linear and Quasilinear Parabolic Problems," Volume I Birkhäuser, Basel, 1995. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [16] A. Friedman, Cancer models and their mathematical analysis, Lect. Notes Math., 1872 (2006), 223-246. doi: 10.1007/11561606_6. [17] D. Gilbarg and T. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, New York, 2001. [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. [19] E. I. Hanzawa, Classical solutions of the Stefan problem, Tôhoku Math. J., 33 (1981), 297-335. [20] T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, Berlin Heidelberg, 1995. [21] A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, Basel, 1995. [22] A. V. Matioc, "Modelling and Analysis of Nonnecrotic Tumors," Südwestdeutcher Verlag für Hochschulschriften, Saarbrücken, 2009. [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.
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