= Lava-crust overshoot under warming
In the <radiogenically heated lava-lake crust model>, initially cool liquid warms exponentially and its <instantaneous stationary lava-crust thickness> $1/(B\theta)$ decreases. The actual crust can cross this moving value, reach a maximum, and melt back to $h_*=1/R$. Overshoot is not automatic. Linearizing about equilibrium gives
$$
v'+SR^2v=-SB(\theta_0-R/B)e^{-B\tau},\qquad v=h-h_*.
$$
If $B<SR^2$ and $\theta_0<R/B$, the slowly decaying forced response has positive sign, so the crust eventually approaches equilibrium from above; an initially absent crust must then have overshot. In contrast, initially hot liquid cools, its instantaneous stationary thickness increases, and an initially absent crust cannot cross it from below: at a proposed first crossing its own <derivative> is zero but the stationary thickness has positive <derivative>. The hot-start crust therefore grows monotonically. These comparisons assume nonnegative liquid superheat.
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