  The dynamics of metric perturbations is explored in the gravity theory with
anomaly-induced quantum corrections. Our first purpose is to derive the
equation for gravitational waves in this theory on the general homogeneous and
isotropic background, and then verify the stability of such background with
respect to metric perturbations. The problem under consideration has several
interesting applications. Our first purpose is to explore the stability of the
classical cosmological solutions in the theory with quantum effects taken into
account. There is an interesting literature about stability of Minkowski and de
Sitter spaces and here we extend the consideration also to the radiation and
matter dominated cosmologies. Furthermore, we analyze the behavior of metric
perturbations during inflationary period, in the stable phase of the Modified
Starobinsky inflation.
