On the viscous Cahn–Hilliard–Oono system with chemotaxis and singular potential

Abstract

We analyze a diffuse interface model that couples a viscous Cahn–Hilliard equation for the phase function φ with a diffusion‐reaction equation for the nutrient concentration σ. The system under consideration also takes into account some important mechanisms like chemotaxis, active transport, and nonlocal interaction of Oono's type. When the spatial dimension is three, we prove the existence and uniqueness of a global weak solution to the model with singular potentials including the physically relevant logarithmic potential. Then we obtain some regularity properties of the weak solution when t > 0. In particular, with the aid of the viscous term ϵ∂tφ, we prove the so‐called instantaneous separation property of the phase variable φ such that it stays away from the pure states ±1 as long as t > 0. Next, we study longtime behavior of the system, by proving the existence of the global attractor in a suitable phase space and characterizing the ω‐limit set. Moreover, we show the existence of an exponential attractor, which implies that the global attractor is of finite fractal dimension.