On a non-local phase-field model for tumour growth with single-well Lennard-Jones potential
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| Format: | Preprint |
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2025
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| _version_ | 1866912273791975424 |
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| author | Grasselli, Maurizio Melzi, Luca Signori, Andrea |
| author_facet | Grasselli, Maurizio Melzi, Luca Signori, Andrea |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_10495 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a non-local phase-field model for tumour growth with single-well Lennard-Jones potential Grasselli, Maurizio Melzi, Luca Signori, Andrea Analysis of PDEs 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. |
| title | On a non-local phase-field model for tumour growth with single-well Lennard-Jones potential |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.10495 |