@inproceedings{875d3400d1dd48a3a3af9d0a2a911627,
title = "On a Diffuse Interface Model for Tumour Growth with Non-local Interactions and Degenerate Mobilities",
abstract = "We study a non-local variant of a diffuse interface model proposed by Hawkins–Daarud et al. (Int. J. Numer. Methods Biomed. Eng. 28:3–24, 2012) for tumour growth in the presence of a chemical species acting as nutrient. The system consists of a Cahn–Hilliard equation coupled to a reaction-diffusion equation. For non-degenerate mobilities and smooth potentials, we derive well-posedness results, which are the non-local analogue of those obtained in Frigeri et al. (European J. Appl. Math. 2015). Furthermore, we establish existence of weak solutions for the case of degenerate mobilities and singular potentials, which serves to confine the order parameter to its physically relevant interval. Due to the non-local nature of the equations, under additional assumptions continuous dependence on initial data can also be shown.",
keywords = "Degenerate mobility, Non-local Cahn–Hilliard equations, Singular potentials, Tumour growth, Weak solutions, Well-posedness",
author = "Sergio Frigeri and Lam, \{Kei Fong\} and Elisabetta Rocca",
note = "This paper is dedicated to Gianni Gilardi, with friendship and admiration. The financial support of the FP7-IDEAS-ERC-StG \#256872 (EntroPhase) and of the project Fondazione Cariplo-Regione Lombardia MEGAsTAR “Matematica d{\textquoteright}Eccellenza in biologia ed ingegneria come acceleratore di una nuova strateGia per l{\textquoteright}ATtRattivit{\`a} dell{\textquoteright}ateneo pavese” is gratefully acknowledged. The paper also benefited from the support of the GNAMPA (Gruppo Nazionale per l{\textquoteright}Analisi Matematica, la Probabilit{\`a} e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) for SF and ER. SF is “titolare di un Assegno di Ricerca dell{\textquoteright}Istituto Nazionale di Alta Matematica (INdAM)” within the project PROGETTO PREMIALE FOE 2014 “Strategic Initiatives for the Environment and Security-SIES”-Metodi variazionali legati al ray-tracing. The authors wish to thank the referee for the careful reading of the paper and useful comments. Publisher Copyright: {\textcopyright} Springer International Publishing AG 2017; INdAM Conference on Optimal Control for Evolutionary PDEs and Related Topics, OCERTO 2016 ; Conference date: 20-06-2016 Through 24-06-2016",
year = "2017",
month = nov,
day = "3",
doi = "10.1007/978-3-319-64489-9\_9",
language = "English",
isbn = "9783319644882",
series = "Springer INdAM Series",
publisher = "Springer Cham",
pages = "217--254",
editor = "Pierluigi Colli and Angelo Favini and Elisabetta Rocca and Giulio Schimperna and J{\"u}rgen Sprekels",
booktitle = "Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs",
edition = "1st",
}