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

Sergio Frigeri, Kei Fong Lam, Andrea Signori

Research output: Contribution to journalArticlepeer-review

Abstract

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.
Original languageEnglish
Pages (from-to)267-308
Number of pages42
JournalEuropean Journal of Applied Mathematics
Volume33
Issue number2
Early online date22 Feb 2021
DOIs
Publication statusPublished - Apr 2022

Scopus Subject Areas

  • Applied Mathematics

User-Defined Keywords

  • Tumour growth
  • degenerate mobility
  • inverse problem
  • non-local Cahn-Hilliard equation
  • singular potentials
  • strong solutions
  • well-posedness
  • Gâteaux differentiability

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