A two-species hyperbolic–parabolic model of tissue growth

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Models of tissue growth are now well established, in particular, in relation to their applications to cancer. They describe the dynamics of cells subject to motion resulting from a pressure gradient generated by the death and birth of cells, itself controlled primarily by pressure through contact inhibition. In the compressible regime, pressure results from the cell densities and when two different populations of cells are considered, a specific difficulty arises; the equation for each cell density carries a hyperbolic character, and the equation for the total cell density has a degenerate parabolic property. For that reason, few a priori estimates are available and discontinuities may occur. Therefore the existence of solutions is a difficult problem. Here, we establish the existence of weak solutions to the model with two cell populations which react similarly to the pressure in terms of their motion but undergo different growth/death rates. In opposition to the method used in the recent paper of Carrillo et al., our strategy is to ignore compactness of the cell densities and to prove strong compactness of the pressure gradient. For that, we propose a new version of Aronson–Bénilan estimate, working in L 2 rather than L ∞ . We improve known results in three directions; we obtain new estimates, we treat dimensions higher than 1 and we deal with singularities resulting from vacuum.