= Explicit solution of gravity-limited frost heave
{title2=$\eta=[1+(\eta_0^{7/4}-1)e^{-7\tau/4}]^{4/7}$}
For $\eta'=\eta^{-3/4}-\eta$, set $u=\eta^{7/4}$ to get $u'=(7/4)(1-u)$. The displayed solution approaches the stable equilibrium $\eta=1$. From the formal zero start it behaves as $(7\tau/4)^{4/7}$ at early times and $1-(4/7)e^{-7\tau/4}$ at late times. The zero-start asymptote is an intermediate reduced-model regime, since the physical thin-film approximation fails at vanishing <ice> thickness.
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