A. Jüngel, Ines Stelzer:

"Entropy structure of a cross-diffusion tumor-growth model";

in: "ASC Report 11/2011", issued by: 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

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