Self-consistent approximate solutions of the semiclassical Einstein equations for a Schwarzschild black hole with its Hawking evaporation

In the present paper, the self-consistent approximate formally analytic solutions of the semiclassical Einstein equations for the equilibrium of a Schwarzschild black hole with its Hawking evaporation are obtained. In the calculation, the approximation is used that the equilibrium system is supposed to consist of a black hole, thermal radiation, and a two-dimensional surface layer. The solution is compared with several related solutions, including York's back-reaction solution.