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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 thecompressible strip forming at the edge in presence of a smooth confiningpotential. Such strip was predicted to host an infinite number of hydrodynamicneutral modes, which however were never observed. A possible explanation may bein their dissipative nature, which was not fully considered before. Heattransport measurements are capable of detecting neutral modes and experiment[H. le Sueur et al., Phys. Rev. Lett. 105, 056803 (2010)] at $\nu = 2$ capturedadditional degrees of freedom transferring heat at the edge. Surprisingly, thebreakdown of heat current quantization has been found. We conjecture that theaforementioned dissipative modes might be responsible for this behavior. Webuild a low-energy effective model and show that the lowest hydrodynamic modecarries 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, amicroscopic model of dissipation is needed to confirm the prediction of thelow-energy approximation.

Heat current in a dissipative quantum Hall edge