  While recent experiments on the spin Seebeck effect have revealed the
decisive role of the magnon contribution to the heat current $Q$ in hybrid
systems containing thin ferromagnetic layers, the available acoustic mismatch
theory does not account for their magnetic properties. Here, we analyze
theoretically the heat transfer through an insulating ferromagnet (F)
sandwiched between two insulators (I). Depending on the relation between the F
thickness, $d$, and the mean free path of phonons generated by magnons,
$l_{ls}$, we reveal two qualitatively different regimes in the nonlinear heat
transport through the F/I interfaces. Namely, in thick F layers the regime of
conventional "Joule" heating with $Q \propto T_s^4$ is realized, in which the
detailed structure of the F/I interfaces is inessential. Here $T_s$ is the
magnon temperature. By contrast, in thin F layers with $d\ll l_{ls}$, most of
phonons emitted by magnons can leave F without being absorbed in its interior,
giving rise to the \emph{magnon overheating} regime with $Q \propto T_s^m$ and
$m\gtrsim7$. Conditions for the examination of both regimes and the
determination of $T_s$ from experiments are discussed. The reported results are
relevant for the theoretical analysis of the spin Seebeck effect and the
development of magnon-based spin caloritronic devices.
