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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 |
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. |
show all references
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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