Well-posedness for a fourth-order nonisothermal tumor growth model of Caginalp type

TL;DR

This paper develops and analyzes a complex mathematical model for tumor growth that includes thermal effects, chemotaxis, and nutrient transport, providing rigorous proofs of well-posedness and solution uniqueness.

Contribution

It introduces a novel nonisothermal phase-field model of Caginalp type for tumor growth, incorporating heat, chemotaxis, and active transport effects, with comprehensive mathematical analysis.

Findings

Existence of weak solutions under certain conditions

Existence of strong solutions with higher regularity

Uniqueness of solutions via continuous dependence

Abstract

We introduce a nonisothermal phase-field system of Caginalp type that describes tumor growth under hyperthermia. The model couples a possibly viscous Cahn-Hilliard equation, governing the evolution of the healthy and tumor phases, with an equation for the heat balance, and a reaction-diffusion equation for the nutrient concentration. The resulting nonlinear system incorporates chemotaxis and active transport effects, and is supplemented with no-flux boundary conditions. The analysis is carried out through a two-step approximation procedure, involving a regularization of the potential and a Faedo-Galerkin discretization scheme. Under stronger regularity assumptions, we further establish the existence of strong solutions and their uniqueness via a continuous dependence result.