= Solution
Let $d$ be an impactor diameter and assume it is much smaller than target diameter $D$, with equal material <mass density>. Neglect <gravitational focusing> and take the encounter speed to be the stated target-size law $v_{\rm rel}=v_pD^p$. The <specific impact energy> is
$$
Q=\frac12\left(\frac dD\right)^3v_p^2D^{2p}.
$$
A <catastrophic planetesimal collision> requires $Q\ge Q_D^*=Q_bD^b$. Therefore the <catastrophic projectile diameter> is
$$
\boxed{D_{\rm cc}=\left(\frac{2Q_b}{v_p^2}\right)^{1/3}D^{1+(b-2p)/3}.}
$$
Use $\max(D_{\rm cc},D_{\min})$ as the actual lower size cutoff when integrating a rate. If this exceeds $D_{\max}$, the specified population contains no destructive impactors. The small-impactor approximation itself requires $D_{\rm cc}\ll D$.
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