We consider the three-dimensional radial Stefan problem which describes the evolution of a radial symmetric ice ball with free boundary \(\begin{aligned} \left\{ \begin{aligned}&\partial _{t}u-\partial _{rr}u-\frac{2}{r}\partial _{r}u=0 \quad in\ r\ge \lambda (t),\\&\partial _{r}u(t,\lambda (t))=-\dot{\lambda }(t),\\&u(t,\lambda (t))=0,\\&u(0,\cdot )=u_{0},\quad \lambda (0)=\lambda _{0}. \end{aligned}\right. \end{aligned}\) We prove the existence in the radial class of finite time melting with rates \(\begin{aligned} \lambda (t)=\left\{ \begin{aligned}&4\sqrt{\pi }\frac{\sqrt{T-t}}{|\log (T-t)|}(1+o_{t\rightarrow T}(1)),\\&c(u_{0},k)(1+o_{t\rightarrow T}(1))(T-t)^{\frac{k+1}{2}},\quad k\in \mathbb {N}^{*}, \end{aligned}\right. \end{aligned}\) which respectively correspond to the fundamental stable melting rate and a sequence of codimension k unstable rates. Our analysis mainly depends on the methods developed in Hadžić and Raphaël (J Eur Math Soc 21(11):3259–3341, 2019) which deals with the similar problems in two dimensions and also the construction of both stable and unstable finite time blow-up solutions for the harmonic heat flow in Raphaël and Schweyer (Anal PDE 7(8):1713–1805, 2014), Raphaël and Schweyer (Commun Pure Appl Math 66(3):414–480, 2013).
