## An existence result to a strongly coupled degenerated system arising in tumor modeling.(English)Zbl 1173.34356

Summary: 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 zero, we prove existence of a global weak solution with finite entropy to the problem by using an approximation procedure and a time discrete scheme.

### MSC:

 34K50 Stochastic functional-differential equations 92C99 Physiological, cellular and medical topics 35K65 Degenerate parabolic equations 35D05 Existence of generalized solutions of PDE (MSC2000)

### Keywords:

global weak solution; finite entropy
Full Text:

### References:

 [1] 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. · Zbl 1012.92017 [2] 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. · Zbl 1334.92190 [3] 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. · Zbl 1040.92015 [4] 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. · Zbl 1046.92023 [5] 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. · Zbl 1015.92020 [6] L. Preziosi, Cancer Modelling and Simulation, Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2003. · Zbl 1039.92022 [7] 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. · Zbl 1109.35358 [8] P. Lauren\ccot 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. · Zbl 1090.35055 [9] 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. · Zbl 1096.35060 [10] 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. · Zbl 1047.35058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.