WIAS Preprint No. 2939, (2022)

Cahn--Hilliard--Brinkman model for tumor growth with possibly singular potentials


  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Gilardi, Gianni
    ORCID: 0000-0002-0651-4307
  • Signori, Andrea
    ORCID: 0000-0001-7025-977X
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604

2020 Mathematics Subject Classification

  • 35K35 35K86 35Q92 35Q35 92C17 92C50


  • Cahn--Hilliard equation, Cahn--Hilliard--Brinkman system, tumor growth model, chemotaxis, singular potential, Dirichlet boundary condition, advection-reaction-diffusion equation




We analyze a phase field model for tumor growth consisting of a Cahn--Hilliard--Brinkman system, ruling the evolution of the tumor mass, coupled with an advection-reaction-diffusion equation for a chemical species acting as a nutrient. The main novelty of the paper concerns the discussion of the existence of weak solutions to the system covering all the meaningful cases for the nonlinear potentials; in particular, the typical choices given by the regular, the logarithmic, and the double obstacle potentials are admitted in our treatise. Compared to previous results related to similar models, we suggest, instead of the classical no-flux condition, a Dirichlet boundary condition for the chemical potential appearing in the Cahn--Hilliard-type equation. Besides, abstract growth conditions for the source terms that may depend on the solution variables are postulated.

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