= Solution
Let $\mathcal I$ be the sphere's mechanical <moment of inertia> about its rotation axis, and let $\omega_i=\omega(0)$. For an undriven rigid sphere the rotational <energy> is $E_{\rm rot}=\mathcal I\omega^2/2$. In the weak-radiation-loss approximation, <conservation of energy> gives
$$
\mathcal I\omega\dot\omega=-\overline P=-K\omega^4,\qquad
\dot\omega=-\frac K{\mathcal I}\omega^3.
$$
Integrating $d(\omega^{-2})/dt=2K/\mathcal I$ gives
$$
\boxed{\omega(t)=\frac{\omega_i}{\sqrt{1+(2K\omega_i^2/\mathcal I)t}}
=\omega_i\left[1+\frac{\mu_0p_0^2\omega_i^2}{3\pi c\mathcal I}t\right]^{-1/2}.}
$$
Equivalently, the characteristic spin-down time is
$$
\boxed{\tau=\frac{\mathcal I}{2K\omega_i^2}
=\frac{27c\mathcal I}{16\pi\mu_0\sigma_0^2R^6\omega_i^2}.}
$$
The <radiative spin-down of a rotating electric dipole> therefore has <angular frequency> that decays as $t^{-1/2}$ at late times within this model, rather than exponentially. The original paper gives no mass or mass-density profile, so $\mathcal I$ cannot be determined from $R$ and the <relative permittivity>. If a homogeneous solid sphere of mass $M$ is additionally assumed, insert $\mathcal I=2MR^2/5$; uniform dielectric response alone does not imply this mechanical mass distribution.
The approximation is important. If the dipole angle is $\phi(t)$ with $\dot\phi=\omega(t)$, direct <differentiation> gives $|\ddot{\mathbf p}|^2=p_0^2(\omega^4+\dot\omega^2)$. Using the constant-frequency radiation law instantaneously neglects the second term and assumes $|\dot\omega|\ll\omega^2$. This gives the standard leading slow spin-down solution above. An exact nonadiabatic <radiation reaction> evolution would require the mechanical and near-field dynamics beyond the information supplied, rather than treating this leading law as exact at arbitrarily rapid damping.
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