WIAS Preprint No. 3234, (2025)

On the hyperbolic relaxation of the chemical potential in a phase field tumor growth model


Authors

  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Rocca, Elisabetta
    ORCID: 0000-0002-9930-907X
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604

2020 Mathematics Subject Classification

  • 35M33 37N25 35K57 35B30 35B65 35B40

Keywords

  • Tumor growth models, singular potentials, hyperbolic relaxation, initial-boundary value problem well-posedness, continuous dependence, regularity, asymptotic convergence

DOI

10.20347/WIAS.PREPRINT.3234

Abstract

In this paper, we study a phase field model for a tumor growth model of Cahn--Hilliard type in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. We show that the resulting initial-boundary value problem is well posed and that its solutions depend continuously on two given functions: one appearing in the mass balance equation and one in the nutrient equation, representing, respectively, sources of drugs (e.g. chemotherapy) and antiangiogenic therapy. We also discuss regularity properties of the solutions. Moreover, in the case of a constant proliferation function, we rigorously analyze the asymptotic behavior as the coefficient of the inertial term tends to zero, establishing convergence to the corresponding viscous Cahn--Hilliard tumor growth model. Our results apply to a broad class of double-well potentials, including nonsmooth ones.

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