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Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition
School of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China |
In this paper we study existence of nonradial stationary solutions of a free boundary problem modeling the growth of nonnecrotic tumors. Unlike the models studied in existing literatures on this topic where boundary value condition for the nutrient concentration $ \sigma $ is linear, in this model this is a nonlinear boundary condition. By using the bifurcation method, we prove that nonradial stationary solutions do exist when the surface tension coefficient $ \gamma $ takes values in small neighborhoods of certain eigenvalues of the linearized problem at the radial stationary solution.
References:
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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. |
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M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
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S. Cui,
Analysis of a free boundary problem modelling tumor growth, Acta Math. Sin. (Engl. Ser.), 21 (2005), 1071-1082.
doi: 10.1007/s10114-004-0483-3. |
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S. Cui and J. Escher,
Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth, Comm. Partial Differential Equations, 33 (2008), 636-655.
doi: 10.1080/03605300701743848. |
| [5] |
S. 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. |
| [6] |
S. Cui and Y. Zhuang,
Bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis, J. Math. Anal. Appl., 468 (2018), 391-405.
doi: 10.1016/j.jmaa.2018.08.022. |
| [7] |
J. Escher and A.-V. Matioc,
Bifurcation analysis for a free boundary problem modeling tumor growth, Arch. Math. (Basel), 97 (2011), 79-90.
doi: 10.1007/s00013-011-0276-8. |
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J. Escher and G. Simonett, Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28 1997, 1028–1047.
doi: 10.1137/S0036141095291919. |
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M. A. Fontelos and A. Friedman,
Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206.
|
| [10] |
A. Friedman and B. Hu,
Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194.
doi: 10.1137/060656292. |
| [11] |
A. Friedman and B. Hu,
Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Rational Mech. Anal., 180 (2006), 293-330.
doi: 10.1007/s00205-005-0408-z. |
| [12] |
A. Friedman and B. Hu,
Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation, J. Math. Anal. Appl., 327 (2007), 643-664.
doi: 10.1016/j.jmaa.2006.04.034. |
| [13] |
A. Friedman and K.-Y. Lam,
Analysis of a free-boundary tumor model with angiogenesis, J. Differential Equations, 259 (2015), 7636-7661.
doi: 10.1016/j.jde.2015.08.032. |
| [14] |
A. Friedman and F. 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), 1587-1634.
doi: 10.1090/S0002-9947-00-02715-X. |
| [15] |
Y. Huang, Z. Zhang and B. Hu,
Bifurcation for a free-boundary tumor model with angiogenesis, Nonlinear Anal. Real World Appl., 35 (2017), 483-502.
doi: 10.1016/j.nonrwa.2016.12.003. |
| [16] |
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, New Jersey, 1971.
|
| [17] |
Z. Wang,
Bifurcation for a free boundary problem modeling tumor growth with inhibitors, Nonlinear Anal. Real World Appl., 19 (2014), 45-53.
doi: 10.1016/j.nonrwa.2014.03.001. |
| [18] |
J. Wu,
Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors, Discrete Contin. Dyn. Syst., 39 (2019), 3399-3411.
doi: 10.3934/dcds.2019140. |
| [19] |
J. Wu and S. Cui,
Bifurcation analysis of a mathematical model for the growth of solid tumors in the presence of external inhibitors, Math. Methods Appl. Sci., 38 (2015), 1813-1823.
doi: 10.1002/mma.3190. |
| [20] |
J. Zheng and S. Cui,
Analysis of a tumor-model free boundary problem with a nonliear boundary condition, J. Math. Anal. Appl., 478 (2019), 806-824.
doi: 10.1016/j.jmaa.2019.05.056. |
| [21] |
F. Zhou and S. Cui,
Bifurcation for a free boundary problem modeling the growth of multi-layer tumors, Nonlinear Anal., 68 (2008), 2128-2145.
doi: 10.1016/j.na.2007.01.036. |
| [22] |
F. Zhou, J. Escher and S. Cui,
Bifurcation for a free boundary problem with surface tension modeling the growth of multi-layer tumors, J. Math. Anal. Appl., 337 (2008), 443-457.
doi: 10.1016/j.jmaa.2007.03.107. |
| [23] |
Y. Zhuang and S. Cui,
Analysis of a free boundary problem modeling the growth of multicell spheroids with angiogenesis, J. Differential Equations, 265 (2018), 620-644.
doi: 10.1016/j.jde.2018.03.005. |
show all references
References:
| [1] |
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. |
| [2] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
| [3] |
S. Cui,
Analysis of a free boundary problem modelling tumor growth, Acta Math. Sin. (Engl. Ser.), 21 (2005), 1071-1082.
doi: 10.1007/s10114-004-0483-3. |
| [4] |
S. Cui and J. Escher,
Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth, Comm. Partial Differential Equations, 33 (2008), 636-655.
doi: 10.1080/03605300701743848. |
| [5] |
S. 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. |
| [6] |
S. Cui and Y. Zhuang,
Bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis, J. Math. Anal. Appl., 468 (2018), 391-405.
doi: 10.1016/j.jmaa.2018.08.022. |
| [7] |
J. Escher and A.-V. Matioc,
Bifurcation analysis for a free boundary problem modeling tumor growth, Arch. Math. (Basel), 97 (2011), 79-90.
doi: 10.1007/s00013-011-0276-8. |
| [8] |
J. Escher and G. Simonett, Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28 1997, 1028–1047.
doi: 10.1137/S0036141095291919. |
| [9] |
M. A. Fontelos and A. Friedman,
Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206.
|
| [10] |
A. Friedman and B. Hu,
Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194.
doi: 10.1137/060656292. |
| [11] |
A. Friedman and B. Hu,
Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Rational Mech. Anal., 180 (2006), 293-330.
doi: 10.1007/s00205-005-0408-z. |
| [12] |
A. Friedman and B. Hu,
Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation, J. Math. Anal. Appl., 327 (2007), 643-664.
doi: 10.1016/j.jmaa.2006.04.034. |
| [13] |
A. Friedman and K.-Y. Lam,
Analysis of a free-boundary tumor model with angiogenesis, J. Differential Equations, 259 (2015), 7636-7661.
doi: 10.1016/j.jde.2015.08.032. |
| [14] |
A. Friedman and F. 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), 1587-1634.
doi: 10.1090/S0002-9947-00-02715-X. |
| [15] |
Y. Huang, Z. Zhang and B. Hu,
Bifurcation for a free-boundary tumor model with angiogenesis, Nonlinear Anal. Real World Appl., 35 (2017), 483-502.
doi: 10.1016/j.nonrwa.2016.12.003. |
| [16] |
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, New Jersey, 1971.
|
| [17] |
Z. Wang,
Bifurcation for a free boundary problem modeling tumor growth with inhibitors, Nonlinear Anal. Real World Appl., 19 (2014), 45-53.
doi: 10.1016/j.nonrwa.2014.03.001. |
| [18] |
J. Wu,
Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors, Discrete Contin. Dyn. Syst., 39 (2019), 3399-3411.
doi: 10.3934/dcds.2019140. |
| [19] |
J. Wu and S. Cui,
Bifurcation analysis of a mathematical model for the growth of solid tumors in the presence of external inhibitors, Math. Methods Appl. Sci., 38 (2015), 1813-1823.
doi: 10.1002/mma.3190. |
| [20] |
J. Zheng and S. Cui,
Analysis of a tumor-model free boundary problem with a nonliear boundary condition, J. Math. Anal. Appl., 478 (2019), 806-824.
doi: 10.1016/j.jmaa.2019.05.056. |
| [21] |
F. Zhou and S. Cui,
Bifurcation for a free boundary problem modeling the growth of multi-layer tumors, Nonlinear Anal., 68 (2008), 2128-2145.
doi: 10.1016/j.na.2007.01.036. |
| [22] |
F. Zhou, J. Escher and S. Cui,
Bifurcation for a free boundary problem with surface tension modeling the growth of multi-layer tumors, J. Math. Anal. Appl., 337 (2008), 443-457.
doi: 10.1016/j.jmaa.2007.03.107. |
| [23] |
Y. Zhuang and S. Cui,
Analysis of a free boundary problem modeling the growth of multicell spheroids with angiogenesis, J. Differential Equations, 265 (2018), 620-644.
doi: 10.1016/j.jde.2018.03.005. |
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