On a diffuse interface model of tumor growth

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 phi nonlinearly coupled with a reaction-diffusion equation for psi, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function p(phi) multiplied by the differences of the chemical potentials for phi and psi The system is equipped with no-flux boundary conditions which give the conservation of the total mass, that is, the spatial average of phi + psi 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.