Convergence to Equilibrium for a Thin-Film Equation on a Cylindrical Surface

Abstract

The degenerate parabolic equation u t + ∂ x [u 3 (u xxx + u x − sin x)] = 0 models the evolution of a thin liquid film on a stationary horizontal cylinder. It is shown here that for each mass there is a unique steady state, given by a droplet hanging from the bottom of the cylinder that meets the dry region with zero contact angle. The droplet minimizes the associated energy functional and attracts all strong solutions that satisfy certain energy and entropy inequalities, including all positive solutions. The distance of solutions from the steady state cannot decay faster than a power law.

Weak Solution of a Doubly Degenerate Parabolic Equation

“…Regarding the long time behavior, Carrillo and Toscani [8] proved the convergence to a self-similar solution for equation (1) with n = 1 and Carlen and Ulusoy [7] gave an upper bound on the distance from the self-similar solution. A similar result on a cylindrical surface was obtained in [5].…”

“…(2) u satisfies the boundary conditions (5) and initial condition (4), (3) u is a weak solution of equation ( 2) in the following sense:…”

Strong Solutions of the Thin Film Equation in Spherical Geometry

“…Regarding the longtime behaviour, Carrillo and Toscani [6] proved the convergence to a self-similar solution for equation (1.3) with n = 1 and Carlen and Ulusoy [5] gave an upper bound on the distance from the self-similar solution. A similar result on a cylindrical surface was obtained in [4].…”

“…where C 1 > 0 is independent of ǫ and δ. Similar to [2, Theorem 4.1, p. 190], using (3.10) and (3.22), we can show that the limit solution u δ is non-negative if n ∈ [1,4) and positive if n 4. Next, letting δ → 0, we get a non-negative strong solution.…”

A mountain pass scenario and heteroclinic orbits for thin-film type equations

“…Convergence to a droplet with zero contact angle for solutions of a thin-film equation on a cylindrical surface has recently been established in [8].…”