Communications in Mathematical Sciences

Volume 16 (2018)

Number 3

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

Pages: 821 – 856



Sergio Frigeri (Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Brescia, Italy)

Kei Fong Lam (Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong)

Elisabetta Rocca (Dipartimento di Matematica, Università di Pavia, Italy; and IMATI - C.N.R., Pavia, Italy)

Giulio Schimperna (Dipartimento di Matematica, Università di Pavia, Italy; and IMATI - C.N.R., Pavia, Italy)


We consider a model describing the evolution of a tumor inside a host tissue in terms of the parameters $\varphi_p , \varphi_d$ (proliferating and necrotic cells, respectively), $u$ (cell velocity) and $n$ (nutrient concentration). The variables $\varphi_p , \varphi_d$ satisfy a vectorial Cahn–Hilliard-type system with nonzero forcing term (implying that their spatial means are not conserved in time), whereas $u$ obeys a variant of Darcy’s law and $n$ satisfies a quasi-static 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 ($\varphi_p , \varphi_d$) is constrained to remain in the range of physically admissible values. On the other hand, in 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 $\varphi_p$ and $\varphi_d$. 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.


tumor growth, nonlinear evolutionary system, Cahn–Hilliard–Darcy system, existence of weak solutions, logarithmic potentials

2010 Mathematics Subject Classification

35D30, 35K57, 35Q35, 35Q92, 76S05, 92B05, 92C17

This research has been performed in the framework of the project Fondazione Cariplo-Regione Lombardia MEGAsTAR “Matematica d’Eccellenza in biologia ed ingegneria come acceleratore di una nuova strateGia per l’ATtRattività dell’ateneo pavese”. The present paper also benefits from the support of the MIUR-PRIN Grant 2015PA5MP7 “Calculus of Variations” for GS, and of the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) for SF, ER, and GS. SF is “titolare di un Assegno di Ricerca dell’Istituto Nazionale di Alta Matematica”.

Received 5 September 2017

Received revised 6 February 2018

Accepted 6 February 2018

Published 30 August 2018