Anomalously high and sharp peaks in the conductance of intrinsic Josephson junctions in Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+delta}$ (Bi2212) mesas have been universally interpreted as superconducting energy gaps, but here we show they are a result of heating. This interpretation follows from a direct comparison to the equilibrium gap, $mathit Delta$, measured in break junctions on similar Bi2212 crystals. As the dissipated power increases with a greater number of junctions in the mesa, the conductance peak abruptly sharpens and its voltage decreases to well below 2$mathit Delta$. This sharpening, found in our experimental data, defies conventional intuition of heating effects on tunneling spectra, but it can be understood as an instability into a nonequilibrium two-phase coexistent state. The measured peak positions occur accurately within the voltage range that an S-shaped backbending is found in the {it calculated} current-voltage curves for spatially {it uniform} self-heating and that S-shape implies the potential for the uniform state to be unstable.
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