On a diffuse interface model of tumour growth

• Published in: European journal of applied mathematics Vol. 26; no. 2; pp. 215 - 243
• Main Authors: FRIGERI, SERGIO, GRASSELLI, MAURIZIO, ROCCA, ELISABETTA
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.04.2015
• Subjects: Applied mathematics; Boundary conditions; Cauchy problem; Conservation; Diffusion; Joining; Mathematical analysis; Mathematical models; Reaction-diffusion equations; Studies; Tumors; Tumours; global attractors; reaction–diffusion equations; Cahn–Hilliard equations; well-posedness; weak solutions; diffuse interface; tumour growth
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792514000436

We consider a diffuse interface model of tumour growth proposed by A. Hawkins-Daruud et al. ((2013) J. Math. Biol.67 1457–1485). This model consists of the Cahn–Hilliard equation for the tumour cell fraction ϕ nonlinearly coupled with a reaction–diffusion equation for ψ, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function p(ϕ) multiplied by the differences of the chemical potentials for ϕ and ψ. The system is equipped with no-flux boundary conditions which give the conservation of the total mass, that is, the spatial average of ϕ + ψ. Here, we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential F and p satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that p satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.