Global and exponential attractors for a Cahn–Hilliard equation with logarithmic potentials and mass source

Abstract

We investigate the long-time behaviour of strong solutions to a generalised Cahn–Hilliard equation with singular logarithmic potentials and a solution-dependent mass source term. Under appropriate choices of the latter, such models have been used for image inpainting and various biological applications. The logarithmic potential is used to ensure the phase field variable stays in the physically relevant interval. Under rather general assumptions on the source term, we first demonstrate a dissipative estimate, leading to uniform-in-time bounds and regularity assertions for previously established weak solutions. Using these, for two spatial dimensions, we prove global strong well-posedness for the model, and demonstrate the existence of the global attractor and exponential attractors with finite fractal dimensions. Moreover, a backwards uniqueness property is shown for the dynamics restricted to the global attractor.