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We consider a (degenerate) cross-diffusion model of tumour growth structured by phenotypic trait. We prove the existence of weak solutions and the incompressible limit as the pressure becomes stiff extending methods recently introduced in the context of two-species cross-diffusion systems. In the stiff-pressure limit, the compressible model generates a free boundary problem of Hele-Shaw type. Moreover, we prove a new  L4 -bound on the pressure gradient.

Phenotypic heterogeneity in a model of tumour growth: existence of solutions and incompressible limit