Global exponential stability of stationary profiles in a thin film equation with second-order diffusion

We study existence and long-time behavior of weak solutions to a thin-film equation with a confinement potential and a second-order degenerate diffusion term. It is known that in absence of second order effects, solutions for general initial data converge at an exponential rate in time to the unique stationary profile. Our main result is that if the strength of the additional forces is sufficiently small, this global exponential equilibration behavior persists, at a slightly smaller rate. Our proof uses the formulation of the equation as a Wasserstein gradient flow, and an auxiliary lower-order Lyapunov functional.