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 $dll l_{ls}$, 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 $mgtrsim7$. 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.
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