On a diffuse interface model of tumor growth

We consider a diffuse interface model of tumor growth proposed by A.~Hawkins-Daruud et al. This model consists of the Cahn-Hilliard equation for the tumor cell fraction $\varphi$ 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 $p(\varphi)$ multiplied by the differences of the chemical potentials for $\varphi$ and $\psi$. The system is equipped with no-flux boundary conditions which entails the conservation of the total mass, that is, the spatial average of $\varphi+\psi$. 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.