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

TL;DR

This paper introduces a hyperbolic relaxation of the chemical potential in a phase field tumor growth model, proving well-posedness, regularity, and asymptotic convergence to classical models, with broad applicability.

Contribution

It presents the first rigorous analysis of a hyperbolic relaxation in a Cahn-Hilliard tumor model, including well-posedness and asymptotic behavior results.

Findings

Well-posedness of the hyperbolic model established

Solutions depend continuously on source functions

Convergence to viscous Cahn-Hilliard model as inertial term vanishes

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.