We consider a mathematical model to describe the growth of a vascular tumor including
tumor cells, macrophages, and blood vessels. The resulting system of equations is reduced to a
strongly $2\times 2$ coupled nonlinear parabolic system of degenerate type. Assuming the initial data
are far enough from 0, we prove existence of a global weak solution with finite entropy to the
problem by using an approximation procedure and a time discrete scheme.
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References
C. J. W. Breward, H. M. Byrne, and C. E. Lewis, ``The role of cell-cell interactions in a two-phase model for avascular tumour growth,'' Journal of Mathematical Biology, vol. 45, no. 2, pp. 125--152, 2002.
C. J. W. Breward, H. M. Byrne, and C. E. Lewis, ``A multiphase model describing vascular tumour growth,'' Bulletin of Mathematical Biology, vol. 65, no. 4, pp. 609--640, 2003.
H. M. Byrne, J. R. King, D. L. S. McElwain, and L. Preziosi, ``A two-phase model of solid tumour growth,'' Applied Mathematics Letters, vol. 16, no. 4, pp. 567--573, 2003.
H. M. Byrne and L. Preziosi, ``Modelling solid tumour growth using the theory of mixtures,'' Mathematical Medicine and Biology, vol. 20, no. 4, pp. 341--366, 2003.
T. L. Jackson and H. M. Byrne, ``A mechanical model of tumor encapsulation and transcapsular spread,'' Mathematical Biosciences, vol. 180, no. 1, pp. 307--328, 2002.
L. Preziosi, Cancer Modelling and Simulation, Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2003.
D. Le, ``Cross diffusion systems on $n$ spatial dimensional domains,'' in Proceedings of the 5th Mississippi State Conference on Differential Equations and Computational Simulations (Mississippi State, MS, 2001), vol. 10 of Electronic Journal of Differential Equations, pp. 193--210, Southwest Texas State University, San Marcos, Tex, USA, 2003.
P. Laurençot and D. Wrzosek, ``A chemotaxis model with threshold density and degenerate diffusion,'' in Nonlinear Elliptic and Parabolic Problems, vol. 64 of Progress in Nonlinear Differential Equations and Their Applications, pp. 273--290, Birkhäuser, Basel, Switzerland, 2005.
L. Chen and A. Jüngel, ``Analysis of a parabolic cross-diffusion population model without self-diffusion,'' Journal of Differential Equations, vol. 224, no. 1, pp. 39--59, 2006.
G. Galiano, A. Jüngel, and J. Velasco, ``A parabolic cross-diffusion system for granular materials,'' SIAM Journal on Mathematical Analysis, vol. 35, no. 3, pp. 561--578, 2003. \endthebibliography
