We investigate the large time behavior of compactly supported smooth solutions for a one-dimensional thin-film equation with linear mobility in the regime of partial wetting. We show the stability of steady state solutions. Relaxation rates are obtained for initial data which are close to a steady state in a suitable sense. The proof uses the Lagrangian coordinates. Our method is to establish and exploit differential relations between the energy and the dissipation as well as some interpolation inequalities. Our result is different from earlier results because here we consider solutions with finite mass.

中文翻译：

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具有部分润湿和线性迁移率的一维薄膜方程的平衡松弛

我们研究了部分润湿状态下具有线性迁移率的一维薄膜方程的紧支撑光滑解的长时间行为。我们展示了稳态解决方案的稳定性。在合适的意义上，获得接近稳态的初始数据的弛豫率。证明使用拉格朗日坐标。我们的方法是建立和利用能量与耗散之间的微分关系以及某些插值不等式。我们的结果与先前的结果不同，因为在这里我们考虑有限质量的解。