On a multi-species Cahn–Hilliard–Darcy tumor growth model with singular potentials

• DOI: 10.4310/cms.2018.v16.n3.a11
• Publication Date: 2018-08-30T14:55:12Z
• Authors: Sergio Frigeri; Kei Fong Lam; Elisabetta Rocca; Giulio Schimperna
• Keywords: 35D30, 35Q35, 35Q92, 35K57, 76S05, 92C17, 92B05; Cahn Hilliard Darcy system; 530; Tumor growth; nonlinear evolutionary system; Cahn-Hilliard-Darcy system; existence of weak solutions; logarithmic potentials; nonlinear evolutionary system; 01 natural sciences; 510; Mathematics - Analysis of PDEs; FOS: Mathematics; existence of weak solutions; logarithmic potentials; Cahn-Hilliard-Darcy system; 0101 mathematics; Tumor growth; Analysis of PDEs (math.AP)

41 pages<br/>We consider a model describing the evolution of a tumor inside a host tissue in terms of the parameters $��_p$, $��_d$ (proliferating and dead cells, respectively), $u$ (cell velocity) and $n$ (nutrient concentration). The variables $��_p$, $��_d$ satisfy a Cahn-Hilliard type system with nonzero forcing term (implying that their spatial means are not conserved in time), whereas $u$ obeys a form of the Darcy law and $n$ satisfies a quasistatic diffusion equation. The main novelty of the present work stands in the fact that we are able to consider a configuration potential of singular type implying that the concentration vector $(��_p,��_d)$ is constrained to remain in the range of physically admissible values. On the other hand, in view of the presence of nonzero forcing terms, this choice gives rise to a number of mathematical difficulties, especially related to the control of the mean values of $��_p$ and $��_d$. For the resulting mathematical problem, by imposing suitable initial-boundary conditions, our main result concerns the existence of weak solutions in a proper regularity class.<br/>