On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects

Harald Garcke, Kei Fong LAM, Andrea Signori*

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

26 Citations (Scopus)


Mechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn–Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell–cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. In the overall model an equation of Cahn–Hilliard type is coupled to the system of linear elasticity and a reaction–diffusion equation for a nutrient concentration. The highly non-linear coupling between a fourth-order Cahn–Hilliard equation and the quasi-static elasticity system lead to new challenges which cannot be dealt within a gradient flow setting which was the method of choice for other elastic Cahn–Hilliard systems. We show existence, uniqueness and regularity results. In addition, several continuous dependence results with respect to different topologies are shown. Some of these results give uniqueness for weak solutions and other results will be helpful for optimal control problems.

Original languageEnglish
Article number103192
JournalNonlinear Analysis: Real World Applications
Early online date18 Jul 2020
Publication statusPublished - Feb 2021

Scopus Subject Areas

  • Analysis
  • Engineering(all)
  • Economics, Econometrics and Finance(all)
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Cahn–Hilliard equation
  • Elliptic–parabolic system
  • Existence and uniqueness
  • Linear elasticity
  • Mechanical effects
  • Tumour growth


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