= Characteristic transformation for conical ice drainage
Putting $y=x^{5/3}$ and $w=x^{1/3}h$ transforms $h_t+x^{-1}(xDh^3)_x=0$ into the <scalar conservation law>
$$
w_t+\frac{5D}{3}\partial_y(w^3)=0.
$$
Along smooth <characteristic curves>, $w$ is constant and $y$ grows at speed $5Dw^2$. Starting from $h_0(s)$ gives
$$
h(x,t)x^{1/3}=h_0(s)s^{1/3},\qquad x^{5/3}=s^{5/3}+5D[h_0(s)s^{1/3}]^2t.
$$
After <characteristic crossing>, the <Rankine-Hugoniot condition> and <entropy solution> select the physical front.
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