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Static black holes in the conformal anomaly-sourced semi-classical GeneralRelativity in four dimensions were extended to rotating, stationary solutions,recently. These quantum-corrected black holes show different features comparedto the Kerr black hole and need for further extensions. Here we remove thecondition of stationarity and find radiating (Kerr-Vaidya-type) solutions inthe same theory augmented with a cosmological constant. As long as the couplingconstant $\alpha$ of the $A$-type trace anomaly is non-zero, we show that $i)$the cosmological constant is bounded from above, i.e $\Lambda \le \frac{3}{4\alpha}$; $ii)$ static black holes exist but they may not be unique; $iii)$static black holes do not satisfy the second law of black hole thermodynamics;$iv)$ static black holes may have unstable inner horizons; $v)$ In thenonstationary and axially symmetric case, stability of the event horizon andthe second law of thermodynamics black holes are problematic.

Kerr-Vaidya type radiating black holes in semi-classical gravity with conformal anomaly