In this paper, we extend the results of [8] by proving exponential asymptotic $H^1$-convergence of solutions to a one-dimensional singular heat equation with $L^2$-source term that describe evolution of viscous thin liquid sheets while considered in the Lagrange coordinates. Furthermore, we extend this asymptotic convergence result to the case of a time inhomogeneous source. This study has also independent interest for the porous medium equation theory.

中文翻译：

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交叉扩散模型中的偏析效应和缺口形成

在本文中，我们通过证明一维奇异热方程的线性渐近$ H ^ 1 $-收敛性与描述粘性薄液体片材演化的$ L ^ 2 $-源项，扩展了[8]的结果而在拉格朗日坐标中考虑。此外，我们将这种渐近收敛结果扩展到时间不均匀源的情况。这项研究对多孔介质方程理论也有独立的兴趣。