On the hyperbolic relaxation of the chemical potential in a phase field tumor growth model
Pierluigi Colli, Elisabetta Rocca, Jürgen Sprekels
TL;DR
The paper studies a diffuse-interface tumor growth model of Cahn–Hilliard type with hyperbolic relaxation of the chemical potential and a viscous CH term, incorporating two distributed therapy controls. It establishes a robust well-posedness theory in a general, possibly nonsmooth potential setting, and proves continuous dependence and higher regularity results leading to strong solutions. In the constant-proliferation case, it rigorously derives a vanishing-inertia limit, proving convergence to a viscous Cahn–Hilliard tumor model with an explicit error rate. These results provide a rigorous mathematical foundation for hyperbolic regularization in tumor-phase-field models and lay groundwork for associated optimal control analyses.
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.