Long-Time Asymptotics for Strong Solutions¶of the Thin Film Equation

In this paper we investigate the large-time behavior of strong solutions to the one-dimensional fourth order degenerate parabolic equation ut=−(uuxxx)x, modeling the evolution of the interface of a spreading droplet. For nonnegative initial values u0(x)∈H1(ℝ), both compactly supported or of finite second moment, we prove explicit and universal algebraic decay in the L1-norm of the strong solution u(x,t) towards the unique (among source type solutions) strong source type solution of the equation with the same mass. The method we use is based on the study of the time decay of the entropy introduced in [13] for the porous medium equation, and uses analogies between the thin film equation and the porous medium equation.