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

Static black holes in the conformal anomaly-sourced semi-classical General Relativity in four dimensions were extended to rotating, stationary solutions, recently. These quantum-corrected black holes show different features compared to the Kerr black hole and need for further extensions. Here we remove the condition of stationarity and find radiating (Kerr-Vaidya-type) solutions in the same theory augmented with a cosmological constant. As long as the coupling constant $\alpha$ 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 the nonstationary and axially symmetric case, stability of the event horizon and the second law of thermodynamics black holes are problematic.