On a diffuse interface model of tumour growth

Abstract

We consider a diffuse interface model of tumour growth proposed by A. Hawkins-Daruudet al.((2013)J. Math. Biol.671457–1485). 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 functionp(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 potentialFandpsatisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided thatpsatisfies 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 ana prioribounded energy.