Paper detail

On the energy-minimizing steady states of a thin film equation

Steady states of the thin film equation $u_t+[u^3 (u_xxx + α^2 u_x -\sin(x) )]_x=0$ are considered on the periodic domain $Ω= (-π,π)$. The equation defines a generalized gradient flow for an energy functional that controls the $H^1$-norm. The main result establishes that there exists for each given mass a unique nonnegative function of minimal energy. This minimizer is symmetric decreasing about $x=0$. For $α<1$ there is a critical value for the mass at which the minimizer has a touchdown zero. If the mass exceeds this value, the minimizer is strictly positive. Otherwise, it is supported on a proper subinterval of the domain and meets the dry region at zero contact angle. A second result explores the relation between strict positivity and exponential convergence for steady states. It is shown that positive minimizers are locally exponentially attractive, while the distance from a steady state with a dry region cannot decay faster than a power law.