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Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion
1. | Department of Applied Mathematics, Dong Hua University, Shanghai 200051 |
2. | Departamento de Matemática Aplicada, E.T.S.I. Sistemas Informáticos, Universidad Politécnica de Madrid, 28031 Madrid, Spain |
References:
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doi: 10.1007/978-3-663-11336-2_1. |
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A. Fasano, M. A. Herrero and M. R. Rodrigo, Slow and fast invasion waves in a model of acid-mediated tumour growth, Math. Biosci., 220 (2009), 45-56.
doi: 10.1016/j.mbs.2009.04.001. |
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R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753. |
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R. A. Gatenby and R. J. Gillies, Why do cancers have high aerobic glycolysis?, Nat. Rev. Cancer, 4 (2004), 891-899. |
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R. J. Gillies, D. Verduzco and R. A. Gatenby, Evolutionary dynamics of cancer and why targeted therapy does not work, Nat. Rev. Cancer, 12 (2012), 487-493. |
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O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl. 23, Providence, RI, 1968. |
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J. D. Murray, Mathematical Biology: I. An Introduction, Interdisciplinary Applied Mathematics, 3rd edn, Springer, New York, 2002. |
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J. B. McGillen, E. A. Gaffney, N. K. Martin and P. K. Maini, A general reaction-diffusion model of acidity in cancer invasion, J. Math. Biol., 68 (2014), 1199-1224.
doi: 10.1007/s00285-013-0665-7. |
[10] |
M. Negreanu and J. I. Tello, On a comparison method to reaction-diffusion systems and its applications to chemotaxis, Discr. Cont. Dyn. Syst. B, 18 (2013), 2669-2688.
doi: 10.3934/dcdsb.2013.18.2669. |
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H. J. Park, J. C. Lyons, T. Ohtsubo and C. W. Song, Acidic environment causes apoptosis by increasing caspase activity, British J. Cancer, 80 (1999), 1892-1897. |
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Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Analysis: RWA, 12 (2011), 418-435.
doi: 10.1016/j.nonrwa.2010.06.027. |
show all references
References:
[1] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in: Schmeisser, Triebel (Eds.), Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Texte zur Mathematik, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[2] |
J. J. Casciari, S. V. Sotirchos and R. M. Sutherland, Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular pH, J. Cell Physiol., 151 (1992), 386-394. |
[3] |
A. Fasano, M. A. Herrero and M. R. Rodrigo, Slow and fast invasion waves in a model of acid-mediated tumour growth, Math. Biosci., 220 (2009), 45-56.
doi: 10.1016/j.mbs.2009.04.001. |
[4] |
R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753. |
[5] |
R. A. Gatenby and R. J. Gillies, Why do cancers have high aerobic glycolysis?, Nat. Rev. Cancer, 4 (2004), 891-899. |
[6] |
R. J. Gillies, D. Verduzco and R. A. Gatenby, Evolutionary dynamics of cancer and why targeted therapy does not work, Nat. Rev. Cancer, 12 (2012), 487-493. |
[7] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl. 23, Providence, RI, 1968. |
[8] |
J. D. Murray, Mathematical Biology: I. An Introduction, Interdisciplinary Applied Mathematics, 3rd edn, Springer, New York, 2002. |
[9] |
J. B. McGillen, E. A. Gaffney, N. K. Martin and P. K. Maini, A general reaction-diffusion model of acidity in cancer invasion, J. Math. Biol., 68 (2014), 1199-1224.
doi: 10.1007/s00285-013-0665-7. |
[10] |
M. Negreanu and J. I. Tello, On a comparison method to reaction-diffusion systems and its applications to chemotaxis, Discr. Cont. Dyn. Syst. B, 18 (2013), 2669-2688.
doi: 10.3934/dcdsb.2013.18.2669. |
[11] |
H. J. Park, J. C. Lyons, T. Ohtsubo and C. W. Song, Acidic environment causes apoptosis by increasing caspase activity, British J. Cancer, 80 (1999), 1892-1897. |
[12] |
Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Analysis: RWA, 12 (2011), 418-435.
doi: 10.1016/j.nonrwa.2010.06.027. |
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