Abstract
This paper deals with a nonlinear system of partial differential equations modeling a simplified tumor-induced angiogenesis taking into account only the interplay between tumor angiogenic factors and endothelial cells. Considered model assumes a nonlinear flux at the tumor boundary and a nonlinear chemotactic response. It is proved that the choice of some key parameters influences the long-time behavior of the system. More precisely, we show the convergence of solutions to different semi-trivial stationary states for different range of parameters.
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Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Schmeisser, H.J., Triebel, H. (eds.) Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Texte zur Mathematik, vol. 133, pp. 9–126 (1993)
Amann H., López-Gómez J.: A priori bounds and multiple solutions for superlinear indefinite elliptic problems. J. Differ. Equ. 146, 336–374 (1998)
Anderson A.R.A., Chaplain M.A.J.: Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60, 857–899 (1998)
Cano-Casanova S., López-Gómez J.: Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems. J. Differ. Equ. 178, 123–211 (2002)
Delgado M., Gayte I., Morales-Rodrigo C., Suárez A.: An angiogenesis model with nonlinear chemotactic response and flux at the tumor boundary. Nonlinear Anal. TMA 72, 330–347 (2010)
Delgado M., Suárez A.: Study of an elliptic system arising from angiogenesis with chemotaxis and flux at the boundary. J. Differ. Equ. 244, 3119–3150 (2008)
Fontelos M., Friedman A., Hu B.: Mathematical analysis of a model for the initiation of angiogenesis. SIAM J. Math. Anal. 33, 1330–1355 (2002)
Henry, D.: Geometric theory of semilinear parabolic equations. Lecture Notes Math. vol. 840, Springer (1981)
Kettemann A., Neuss-Radu M.: Derivation and analysis of a system modeling the chemotactic movement of hematopoietic stem cells. J. Math. Biol. 56, 579–610 (2008)
Levine H.A., Sleeman B.D., Nilsen-Hamilton N.: Mathematical modeling of the onset of capillary formation initiating angiogenesis. J. Math. Biol. 42, 195–238 (2001)
López-Gómez J.: The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems. J. Differ. Equ. 127, 263–294 (1996)
Mantzaris N.V., Webb S., Othmer H.G.: Mathematical modeling of tumor-induced angiogenesis. J. Math. Biol. 49, 183–217 (2004)
Tartar, L.: An introduction to Sobolev spaces and interpolation spaces. Lecture Notes of the Unione Matematica Italiana, vol. 3, Springer (2007)
Umezu K.: Nonlinear elliptic boundary value problems suggested by fermentation. Nonlinear Differ. Equ. Appl. 7, 143–155 (2000)
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Cieślak, T., Morales-Rodrigo, C. Long-time behavior of an angiogenesis model with flux at the tumor boundary. Z. Angew. Math. Phys. 64, 1625–1641 (2013). https://doi.org/10.1007/s00033-013-0302-8
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DOI: https://doi.org/10.1007/s00033-013-0302-8