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On a Diffuse Interface Model for Tumour Growth with Non-local Interactions and Degenerate Mobilities

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Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs

Part of the book series: Springer INdAM Series ((SINDAMS,volume 22))

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.

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Acknowledgements

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’Eccellenza in biologia ed ingegneria come acceleratore di una nuova strateGia per l’ATtRattività dell’ateneo pavese” is gratefully acknowledged. The paper also benefited from the support of the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) for SF and ER. SF is “titolare di un Assegno di Ricerca dell’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.

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Authors and Affiliations

  1. Dipartimento di Matematica e Fisica “N. Tartaglia”, Università Cattolica del Sacro Cuore (Brescia), Via Musei 41, 25121, Brescia, Italy

    Sergio Frigeri

  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Kei Fong Lam

  3. Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100, Pavia, Italy

    Elisabetta Rocca

  4. IMATI-C.N.R., Via Ferrata 5, 27100, Pavia, Italy

    Elisabetta Rocca

Authors
  1. Sergio Frigeri
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  2. Kei Fong Lam
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  3. Elisabetta Rocca
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Corresponding author

Correspondence to Sergio Frigeri .

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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Pierluigi Colli

  2. Dipartimento di Matematica, Università di Bologna , Bologna, Italy

    Angelo Favini

  3. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Elisabetta Rocca

  4. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Giulio Schimperna

  5. für Angewandte Analysis und Stochastik (WIAS), Weierstraß-Institut , Berlin, Germany

    Jürgen Sprekels

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Frigeri, S., Lam, K.F., Rocca, E. (2017). On a Diffuse Interface Model for Tumour Growth with Non-local Interactions and Degenerate Mobilities. In: Colli, P., Favini, A., Rocca, E., Schimperna, G., Sprekels, J. (eds) Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs. Springer INdAM Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-64489-9_9

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