On a non-local phase-field model for tumour growth with single-well Lennard-Jones potential

• Published in: Nonlinear analysis: real world applications Vol. 88; p. 104466
• Main Authors: Grasselli, Maurizio, Melzi, Luca, Signori, Andrea
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.04.2026
• Subjects: Cahn–Hilliard equation; Continuous dependence on the initial datum; Lennard-Jones potential; Non-local interactions; Phase-field model; Tumour growth; Viscous relaxation; Weak and strong solutions; Phase-field model; Weak and strong solutions; Continuous dependence on the initial datum; Non-local interactions; 92C17; Viscous relaxation; 35K55; Cahn–Hilliard equation; Lennard-Jones potential; 35Q92; 92C50; Tumour growth
• ISSN: 1468-1218
• DOI: 10.1016/j.nonrwa.2025.104466

In the present work, we develop a comprehensive and rigorous analytical framework for a non-local phase-field model that describes tumour growth dynamics. The model is derived by coupling a non-local Cahn–Hilliard equation with a parabolic reaction–diffusion equation, which accounts for both phase segregation and nutrient diffusion. Previous studies have only considered symmetric potentials for similar models. However, in the biological context of cell-to-cell adhesion, single-well potentials, like the so-called Lennard-Jones potential, seem physically more appropriate. The Cahn–Hilliard equation with this kind of potential has already been analysed. Here, we take a step forward and consider a more refined model. First, we analyse the model with a viscous relaxation term in the chemical potential and subject to suitable initial and boundary conditions. We prove the existence of solutions, a separation property for the phase variable, and a continuous dependence estimate with respect to the initial data. Finally, via an asymptotic analysis, we recover the existence of a weak solution to the initial and boundary value problem without viscosity, provided that the chemotactic sensitivity is small enough.