  In nonmagnetic insulators, phonons are the carriers of heat. If heat enters
in a region and temperature is measured at a point within phonon mean free
paths of the heated region, ballistic propagation causes a nonlocal relation
between local temperature and heat insertion. This paper focusses on the
solution of the exact Peierls-Boltzmann equation (PBE), the relaxation time
approximation (RTA), and the definition of local temperature needed in both
cases. The concept of a non-local "thermal susceptibility" (analogous to charge
susceptibility) is defined. A formal solution is obtained for heating with a
single Fourier component $P(\vec{r},t)=P_0 \exp(i\vec{k}\cdot\vec{r}-i\omega
t)$, where $P$ is the local rate of heating). The results are illustrated by
Debye model calculations in RTA for a three-dimensional periodic system where
heat is added and removed with $P(\vec{r},t)=P(x)$ from isolated evenly spaced
segments with period $L$ in $x$. The ratio $L/\ell_{\rm min}$ is varied from 6
to $\infty$, where $\ell_{\rm min}$ is the minimum mean free path. The Debye
phonons are assumed to scatter anharmonically with mean free paths varying as
$\ell_{\rm min}(q_D/q)^2$ where $q_D$ is the Debye wavevector. The results
illustrate the expected local (diffusive) response for $\ell_{\rm min}\ll L$,
and a diffusive to ballistic crossover as $\ell_{\rm min}$ increases toward the
scale $L$. The results also illustrate the confusing problem of temperature
definition. This confusion is not present in the exact treatment but is
fundamental in RTA.
