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Strong well-posedness and inverse identification problem of a non-local phase field tumour model with degenerate mobilities

Part of: Physiological, cellular and medical topics Existence theories Parabolic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  22 February 2021

SERGIO FRIGERI ,
KEI FONG LAM  and
ANDREA SIGNORI Open the ORCID record for ANDREA SIGNORI [Opens in a new window]
SERGIO FRIGERI
Affiliation:
Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano via Saldini 50, I-20133Milano, Italy, e-mail: sergio.frigeri@unimi.it
KEI FONG LAM
Affiliation:
Department of Mathematics, Hong Kong Baptist University Kowloon Tong, Hong Kong, e-mail: akflam@hkbu.edu.hk
ANDREA SIGNORI
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca via Cozzi 55, 20125Milano, Italy, e-mail: andrea.signori02@universitadipavia.it
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Abstract

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We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol. 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al. (2012, Int. J. Numer. Method Biomed. Engng.28, 3–24). The model consists of a non-local Cahn–Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction–diffusion equation for the concentration of a nutrient. We prove the existence of strong solutions to the model and establish some high-order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions. Then, we apply the new regularity results to study an inverse problem identifying the initial tumour distribution from measurements at the terminal time. Formulating the Tikhonov regularised inverse problem as a constrained minimisation problem, we establish the existence of minimisers and derive first-order necessary optimality conditions.

Keywords

Tumour growthnon-local Cahn–Hilliard equationdegenerate mobilitysingular potentialsstrong solutionswell-posednessinverse problemGâteaux differentiability

MSC classification

Primary: 35K86: Nonlinear parabolic unilateral problems and nonlinear parabolic variational inequalities 35K61: Nonlinear initial-boundary value problems for nonlinear parabolic equations 49J20: Optimal control problems involving partial differential equations
Secondary: 35Q92: PDEs in connection with biology and other natural sciences 92C50: Medical applications (general)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 2 , April 2022 , pp. 267 - 308
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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