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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.13175 |
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Table of Contents:
- We consider the interior Stefan problem under radial symmetry in two dimension. A water ball surrounded by ice undergoes melting or freezing. We construct a discrete family of global-in-time solutions, both melting and freezing scenarios. The evolution of the free boundary, represented by the radius of the water ball, $λ(t)$ exhibits exponential convergence to a limiting radius value $λ_\infty > 0$, characterized by the asymptotic expression \[ λ(t) = λ_\infty + (1 - λ_\infty)\, e^{-\frac{λ_k}{λ_\infty^2} t + o_{t \to \infty}(1)}, \] where $λ_k$ stands for the $k$-th Dirichlet eigenvalue of the Laplacian on the unit disk for any $k\in \mathbb{N}$. Our approach draws inspiration from the research conducted by Hadžić and Raphaël [24] concerning the exterior radial Stefan problem, which involves an ice ball is surrounded by water. In contrast, the bounded geometry in our setting leads to scenario results in a non-degenerate spectrum, leading to distinctly different long-term behavior. These solutions for each $k$ remain stable under perturbations of co-dimension $k - 1$.