Non-vacuum solutions for evaporating black holes

[Tomas Vencl]

Although the complete quantum gravity is unknown as the exact details of black hole evaporating, is there known some symmetric non vacuum solution of E. equations which includes radiating of matter from central mass ? One can say, that Schwarzschild solution with small perturbation is good enough (as the Hawking radiation is weak), but if BH evaporates, exact solution can not be vacuum and static (which can be connected with extremely different behavior at strong field regions) .
Thanks for some links to learn more about such a solutions .

 

[martinbn]

https://en.wikipedia.org/wiki/Vaidya_metric

 

[Tomas Vencl]

Thanks.
I think undertsand the deriving Eddington-Finkelstein coordinates (eq. 1-5), then at the equation 6 they put M=constant to M(u).
But I do not understand how can be still correct derivative of equation 2 (and next steps) when now M is function u ?
Sorry for this stupid question.