  We explore the heat current in the quantum Hall edge at filling factors $\nu
= 1$ and $\nu = 2$ in the presence of dissipation. Dissipation arises in the
compressible strip forming at the edge in presence of a smooth confining
potential. Such strip was predicted to host an infinite number of hydrodynamic
neutral modes, which however were never observed. A possible explanation may be
in their dissipative nature, which was not fully considered before. Heat
transport measurements are capable of detecting neutral modes and experiment
[H. le Sueur et al., Phys. Rev. Lett. 105, 056803 (2010)] at $\nu = 2$ captured
additional degrees of freedom transferring heat at the edge. Surprisingly, the
breakdown of heat current quantization has been found. We conjecture that the
aforementioned dissipative modes might be responsible for this behavior. We
build a low-energy effective model and show that the lowest hydrodynamic mode
carries a portion of the heat flux quantum which is the same both at $\nu = 1$
and $\nu = 2$. Although our results are consistent with the experiment, a
microscopic model of dissipation is needed to confirm the prediction of the
low-energy approximation.
