Melting and freezing rates of the radial interior Stefan problem in two dimension

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, $\lambda (t)$ exhibits exponential convergence to a limiting radius value ${\lambda }_{\mathrm{\infty }}>0$, characterized by the asymptotic expression $$\lambda (t)={\lambda }_{\mathrm{\infty }}+(1-{\lambda }_{\mathrm{\infty }})e^{-\frac{{\lambda }_k}{{\lambda }_{\mathrm{\infty }}^2}t+o_{t\to \mathrm{\infty }}(1)},$$ where ${\lambda }_k$ 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 $k-1$.