Long-time behaviour of solutions to a singular heat equation with an application to hydrodynamics

TL;DR

This paper proves exponential convergence of solutions to a singular heat equation with source terms, relevant for viscous liquid sheets and porous medium equations, extending previous results to time-inhomogeneous cases.

Contribution

It extends existing convergence results of a singular heat equation to include time-inhomogeneous sources and broader applications in fluid dynamics and porous media.

Findings

Exponential $H^1$-convergence of solutions.

Extension to time-inhomogeneous source terms.

Relevance to viscous thin liquid sheets and porous medium equations.

Abstract

In this paper, we extend the results of [1] 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.