On a relaxed Cahn-Hilliard tumour growth model with single-well potential and degenerate mobility

We consider a phase-field system modelling solid tumour growth. This system consists of a Cahn-Hilliard equation coupled with a nutrient equation. The former is characterised by a degenerate mobility and a singular potential. Both equations are subject to suitable reaction terms which model proliferation and nutrient consumption. Chemotactic effects are also taken into account. Adding an elliptic regularisation, depending on a relaxation parameter $δ>0$, in the equation for the chemical potential, we prove the existence of a weak solution to an initial and boundary value problem for the relaxed system. Then, we let $δ$ go to zero, and we recover the existence of a weak solution to the original system.

💡 Research Summary

The paper studies a phase‑field model for solid tumour growth that couples a Cahn‑Hilliard equation with a nutrient diffusion equation. The phase variable ϕ denotes the tumour cell volume fraction (ϕ=0 healthy tissue, ϕ=1 tumour), while σ represents a key nutrient such as oxygen or glucose. The model incorporates three challenging features: (i) a degenerate mobility b(ϕ)=ϕ(1−ϕ)² that vanishes at the pure phases, (ii) a singular single‑well potential Ψ(ϕ) of Lennard‑Jones type, which is concave near ϕ=0 and convex with a logarithmic singularity as ϕ→1, and (iii) chemotactic cross‑diffusion terms weighted by a non‑negative parameter χ.

To overcome the analytical difficulties posed by the combination of degeneracy and singularity, the authors introduce a relaxation parameter δ>0 and add an elliptic regularisation term –δΔμ to the definition of the chemical potential μ. This transforms the original fourth‑order degenerate Cahn‑Hilliard equation into a coupled system of a parabolic equation for ϕ and an elliptic equation for μ. The potential is split into convex (Ψ⁺) and concave (Ψ⁻) parts, with the singularity confined to Ψ⁺, while Ψ⁻ is extended smoothly to the whole real line. The proliferation function P(ϕ) is chosen to be non‑negative, degenerate at the pure phases, and proportional to the mobility, ensuring compatibility with the entropy structure.

The paper’s first main result (Theorem 2.3) establishes the existence of a weak solution (ϕ_δ, μ_δ, σ_δ) for any fixed δ>0. The solution satisfies

• ϕ_δ ∈ H¹(0,T;V*) ∩ L²(0,T;V),
• μ_δ ∈ L²(0,T;V),
• ϕ_δ−δγμ_δ ∈ H¹(0,T;V*) ∩ L^∞(0,T;V) ∩ L²(0,T;W),
• σ_δ ∈ H¹(0,T;V*) ∩ L^∞(0,T;H) ∩ L²(0,T;V), with the physical bounds 0 ≤ ϕ_δ ≤ 1 a.e. The proof relies on a careful Galerkin approximation combined with regularisations of b, Ψ, and P, and on uniform a‑priori estimates derived from an energy‑entropy inequality that is independent of δ.

The second main result (Theorem 2.8) concerns the limit δ→0. Using compactness arguments and the uniform estimates, the authors pass to the limit in the relaxed system and recover a weak solution of the original, non‑regularised model: ∂ₜϕ − div(b(ϕ)∇μ) = P(ϕ)(σ + χ(1−ϕ) − μ), μ = −γΔϕ + Ψ′(ϕ) − χσ, ∂ₜσ − Δ(σ + χ(1−ϕ)) = −P(ϕ)(σ + χ(1−ϕ) − μ), with homogeneous Neumann boundary conditions and the same initial data. This passage from a non‑local (μ defined through an elliptic problem) to a local formulation is non‑trivial because the degenerate mobility prevents standard compactness; the authors overcome this by exploiting the entropy density η defined via b(s)η″(s)=1 and by establishing strong convergence of the product b(ϕ_δ)∇μ_δ.

The paper also discusses the physical relevance of the model. The degenerate mobility enforces the natural constraint 0≤ϕ≤1, while the singular single‑well potential captures the saturation effect as the tumour phase approaches full occupancy. Chemotaxis (χ>0) drives tumour cells toward nutrient‑rich regions, leading to realistic pattern formation such as branching. The relaxation term δ provides a numerically stable approximation, as shown in earlier works, and the rigorous analysis validates its use.

In summary, the authors deliver the first rigorous existence theory for a Cahn‑Hilliard tumour growth model that simultaneously features degenerate mobility, a singular single‑well potential, and chemotactic cross‑diffusion. Their methodology—potential splitting, entropy‑based estimates, and a δ‑regularisation—offers a robust framework that can be extended to other non‑local phase‑field models and to optimal control or long‑time behaviour analyses.