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.
On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities / S. Frigeri, K.F. Lam, E. Rocca (SPRINGER INDAM SERIES). - In: Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs : In Honour of Prof. Gianni Gilardi / [a cura di] P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels. - [s.l] : Springer, 2017. - ISBN 9783319644882. - pp. 217-254 (( convegno INdAM Conference on Optimal Control for Evolutionary PDEs and Related Topics tenutosi a Cortona nel 2016 [10.1007/978-3-319-64489-9_9].
On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities
S. Frigeri;
2017
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.File | Dimensione | Formato | |
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