Stable localized patterns in thin liquid films

We study a two-dimensional nonlinear evolution equation which describes the three-dimensional spatiotemporal behavior of the air-liquid interface of a thin liquid film lying on the underside of a cooled horizontal plate. We show that the equation has a Liapunov functional and exploit this fact to demonstrate that the Marangoni effect can stabilize the destabilizing effect of gravity (the Rayleigh-Taylor instability) allowing for the existence of stable localized axisymmetric solutions for a wide range of parameter values. Various properties of these structures are discussed.