On a generalization of the Cahn-Hilliard type equation with logarithmic nonlinearities for formation of islands

In this paper, main objective is to demonstrate the existence of solutions for an equation resembling the Cahn-Hilliard model, featuring a proliferation term and a logarithmic nonlinear term. This equation has been conceptualized within the context of interactions in liquid-gas systems, particularly in the context of island formation. The primary challenge lies in the departure from the original Cahn-Hilliard equation, as we no longer maintain conservation of the spatial average mean of the order parameter. This departure introduces the complexity in establishing uniform estimates for the solutions of the approximated problems concerning the regularization parameter, as it may potentially result in finite-time blow-up.