1. The problem asks about the asymptotic stability of the free boundary in the radial Stefan problem, which is a classical moving boundary problem describing phase changes like melting or freezing. 2. The Stefan problem involves solving the heat equation with a moving boundary whose position is part of the solution. The free boundary evolves according to the heat flux and latent heat conditions. 3. The radial Stefan problem assumes spherical symmetry, reducing the problem to one spatial dimension with radius $r$. 4. The asymptotic stability of the free boundary means that small perturbations of the initial free boundary position decay over time, and the solution approaches a steady state or self-similar profile. 5. The key formula involves the heat equation in radial coordinates: $$\frac{\partial u}{\partial t} = \frac{1}{r^{n-1}} \frac{\partial}{\partial r} \left(r^{n-1} \frac{\partial u}{\partial r}\right)$$ where $n$ is the spatial dimension (e.g., $n=3$ for 3D). 6. The free boundary $s(t)$ satisfies the Stefan condition: $$L \frac{ds}{dt} = -k \left. \frac{\partial u}{\partial r} \right|_{r=s(t)^-} + k \left. \frac{\partial u}{\partial r} \right|_{r=s(t)^+}$$ where $L$ is latent heat and $k$ thermal conductivity. 7. Stability analysis typically linearizes the problem around a steady or self-similar solution and studies the eigenvalues of the linearized operator. 8. Results in literature show that under suitable conditions on initial data and parameters, the free boundary is asymptotically stable, meaning perturbations decay exponentially or algebraically in time. 9. Intuitively, heat diffusion smooths out irregularities, and the latent heat condition controls the boundary motion, leading to stability. 10. Therefore, the free boundary of the radial Stefan problem is asymptotically stable under typical physical assumptions. Final answer: The free boundary in the radial Stefan problem is asymptotically stable, meaning small perturbations decay over time and the boundary approaches a steady or self-similar state.