A linear sixth-order partial differential equation (PDE) of ``parabolic'' type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order Sturm--Liouville eigenvalue value problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed trough, and the eigenfunctions form a complete orthonormal set. Using these eigenfunctions, we derive the Green's function for the governing sixth-order PDE on a finite interval and compare it to the known infinite-line solution. Further, we propose a Galerkin spectral method based on the constructed sixth-order eigenfunctions and their derivative expansions. The system of ordinary differential equations for the time-dependent expansion coefficients is solved by standard numerical methods. The numerical approach is applied to versions of the governing PDE with a second-order derivative (in addition to the sixth-order one), which arises from gravity acting on the film. In the absence of gravity, we demonstrate the self-similar intermediate asymptotics of initially localized disturbances on the film surface, at least until the disturbances ``feel'' the finite boundaries, and show that the derived Green's function is the global attractor for such solutions. In the presence of gravity, we use the proposed spectral numerical method to demonstrate that self-similar behavior persists, albeit for shortened intervals of time, even for large values of the gravity-to-bending ratio.

翻译：一类线性六阶“抛物型”偏微分方程描述了在弹性弯曲阻力作用下，当液膜偏离平衡高度的扰动较小时，弹性表面下薄液膜的动力学行为。在有界区域上，对应于封闭槽中薄液膜边界条件的六阶Sturm-Liouville特征值问题是自伴的，且特征函数构成完备的标准正交基。利用这些特征函数，我们推导了有限区间上该六阶偏微分方程的控制格林函数，并将其与已知的无限直线解进行了比较。此外，我们提出了一种基于所构造的六阶特征函数及其导数展开的Galerkin谱方法。时间依赖展开系数对应的常微分方程组通过标准数值方法求解。该数值方法被应用于控制方程中带有二阶导数项（除了六阶项之外）的情形，该二阶项源于重力对液膜的作用。在无重力情况下，我们证明了液膜表面初始局部扰动的自相似中间渐近性（至少在扰动“感知”到有限边界之前），并表明所推导的格林函数是此类解的全局吸引子。在存在重力的情况下，我们利用所提出的谱数值方法证明了即使重力与弯曲力之比很大，自相似行为仍然存在，尽管其持续时间有所缩短。