Global weak solvability, Continuous dependence on data and large time growth of swelling moving interfaces

TL;DR

This paper establishes global existence, uniqueness, and continuous dependence of solutions for a one-dimensional free boundary problem modeling swelling, and analyzes the large-time behavior of the moving interface.

Contribution

It provides the first global weak solvability and stability results for this class of swelling interface problems in one dimension.

Findings

Solutions are globally existent and unique.

The moving boundary never disappears over time.

Some estimates are uniform in time, enabling large-time analysis.

Abstract

We prove a global existence result for weak solutions to a one-dimensional free boundary problem with flux boundary conditions describing swelling along a halfline. Additionally, we show that solutions are not only unique but also depend continuously on data and parameters. The key observation is that the structure of our system of partial differential equations allows us to show that the moving a priori unknown interface never disappears. As main ingredients of the global existence proof, we rely on a local weak solvability result for our problem, uniform estimates of the solution, integral estimates on quantities defined at the free boundary, as well as a fine pointwise lower bound for the position of the moving boundary. Some of the estimates are time-independent. They allow us to explore the large time behavior of the position of the moving boundary. The approach is specific to one-dimensional settings.