Mathematical Biosciences and Engineering (MBE)

Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion
Pages: 193 - 207, Issue 1, February 2016

doi:10.3934/mbe.2016.13.193      Abstract        References        Full text (376.2K)           Related Articles

Youshan Tao - Department of Applied Mathematics, Dong Hua University, Shanghai 200051, China (email)
J. Ignacio Tello - Departamento de Matemática Aplicada, E.T.S.I. Sistemas Informáticos, Universidad Politécnica de Madrid, 28031 Madrid, Spain (email)

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.       
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.       
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.       
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.       
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.       

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