  We consider the three-dimensional radial Stefan problem which describes the
evolution of a radial symmetric ice ball with free boundary
  \begin{equation*}
  \left\{\begin{aligned}
  &\partial_{t}u-\partial_{rr}u-\frac{2}{r}\partial_{r}u=0 \quad in\
r\geq\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{equation*}
  We prove the existence in the radial class of finite time melting with rates
\begin{equation*}
  \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{equation*}
  which respectively correspond to the fundamental stable melting rate and a
sequence of codimension $k$ unstable rates. Our analysis mainly depend on the
methods developed in [17] 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 [49],[50].
