A two species hyperbolic-parabolic model of tissue growth

Piotr Gwiazda (1) , Benoît Perthame (2) , Agnieszká Swierczewska-Gwiazda

1 Institute of Mathematics [Warsaw]
2 LJLL - Laboratoire Jacques-Louis Lions

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 we consider, when pressure results from the cell densities and when two different populations of cells are considered, a specific difficulty arises from the hyperbolic character of the equation for each cell density, and to the parabolic aspect of the equation for the total cell density. 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 [16], our strategy is to ignore compactness on the cell densities and to prove strong compactness on the pressure gradient. We improve known results in two directions; we obtain new estimates, we treat higher dimension than 1 and we deal with singularities resulting from vacuum.

Mots clés: Darcy's law; Cross-diffusion systems; Reaction-diffusion; Hyperbolic-parabolic systems; Mathematical biology

Domaines: Equations aux dérivées partielles [math.AP]

Piotr Gwiazda, Benoît Perthame, Agnieszká Swierczewska-Gwiazda. A two species hyperbolic-parabolic model of tissue growth. Communications in Partial Differential Equations, 2019, 44 (12), pp.1605-1618. ⟨10.1080/03605302.2019.1650064⟩. ⟨hal-01870045⟩