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A. Jüngel, Ines Stelzer:
"Entropy structure of a cross-diffusion tumor-growth model";
Mathematical Models & Methods in Applied Sciences, 22 (2012), 1 - 26.

The mechanical tumor-growth model of Jackson and Byrne is analyzed. The model
consists of nonlinear parabolic cross-diffusion equations in one space dimension for the
volume fractions of the tumor cells and the extracellular matrix (ECM). It describes
tumor encapsulation influenced by a cell-induced pressure coefficient. The global-in-time
existence of bounded weak solutions to the initial-boundary-value problem is proved
when the cell-induced pressure coefficient is smaller than a certain explicit critical value.
Moreover, when the production rates vanish, the volume fractions converge exponentially
fast to the homogeneous steady state. The proofs are based on the existence of entropy
variables, which allows for a proof of the non-negativity and boundedness of the volume
fractions, and of an entropy functional, which yields gradient estimates and provides a
new thermodynamic structure. Numerical experiments using the entropy formulation of
the model indicate that the solutions exist globally in time also for cell-induced pressure
coefficients larger than the critical value. For such coefficients, a peak in the ECM volume
fraction forms and the entropy production density can be locally negative.

Siehe englisches Abstract.

Tumor growth; encapsulation; cross diffusion