Asymmetric travelling waves for the thin film equation (Q488674)

The authors consider the degenerate parabolic equation that describes the evolution of the height of a liquid film spreading over a solid surface, that is, \[ h_t+(h^nh_{xxx})_x=0, \] where \(n\in\mathbb R^+\) is related to the slip condition at the liquid-solid interface. Of particular interest for this equation is a mathematical description of a dewetting film after rupture. In this paper, a free boundary problem is formulated for a traveling wave equation that describes the growth of dewetted regions in the film, while the fluid removed from dewetted regions forms a ridge profile which propagates away, that is, \[ \phi^{n-1}\phi^{\prime\prime\prime}=1,\,\,\, \phi(\eta)\geq 0, \] \[ \phi(0)=\phi(d)=0,\,\,\, 0\leq \eta\leq d. \] Here, \(d>0\) is a free parameter that is fixed by the zero contact angle boundary condition \(\phi^{\prime}(d)=0\). The following results are obtained. The existence and uniqueness of traveling wave solutions to this problem for \(0<n<3\) is shown using the Schauder fixed-point theorem. Lower and upper bounds for the traveling waves are obtained. Non-existence of the compactly supported traveling waves is shown for \(n=3\). The uniqueness of \(d\) is proved for \(n=0\), \(1\), \(2\). The behavior of \(d\) as \(n\nearrow 3\) is studied. Energy dissipation by a class of non-stationary traveling waves is investigated using energy-entropy a priori bounds. The latter result is used to obtain conditions for non-asymptotic stability of the traveling waves.