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A. Jüngel, Ines Stelzer:
"Entropy structure of a cross-diffusion tumor-growth model";
in: "ASC Report 11/2011", herausgegeben von: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2011, ISBN: 978-3-902627-04-9.

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 nonnegativity 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.

http://www.asc.tuwien.ac.at/preprint/2011/asc11x2011.pdf