  Aim of the paper is to obtain 2d analogue of the backreaction equation which
will be useful to study final state of quantum perturbed spherically symmetric
curved space times. Thus we take Einstein-massless-scalar $\psi$ tensor gravity
model described on class of spherically symmetric curved space times. We
rewrite the action functional in 2d analogue in terms of dimensionless
dilaton-matter field $(\chi=\Phi\psi)$ where dilaton field $\Phi$ is conformal
factor of 2-sphere. Then we seek renormalized expectation value of quantum
dilaton-matter field stress tensor operator by applying Hadamard
rennormalization prescription. Singularity of the Green function is assumed to
be has logarithmic form. Covariantly conservation condition on the renormalized
quantum dilaton-matter stress tensor demands to input a variable cosmological
parameter $\lambda(x)$. Energy conditions (weak, strong and null) is studied on
the obtained renormalized stress tensor leading to dynamical equations for
$\lambda(x), \Phi$ and quantum vacuum state $W_0(x)=<0|\hat{\chi}^2|0>_{ren}.$
In weak quantum field limits our obtained trace anomaly corresponds to one
which obtained from zeta function regularization method. Setting null-like
apparent horizon equation $\nabla_c\Phi\nabla^c\Phi=0,$ our procedure predicts
that physically correct value of the parameter in the anomaly trace
$\frac{1}{24\pi}\{R-\alpha\frac{\nabla_{c}\nabla^{c}\Phi}{\Phi}+(\alpha-6)\frac{\nabla_c\Phi\nabla^c\Phi}{\Phi^2}
\}$ should be $\alpha=6.$ At last we solved the backreaction equation and
obtained explicitly metric field solution in the slow varying limits of the
quantum and dilaton fields which has black holes topology and its singularity
is covered by apparent horizon hypersurface.
