On decay of solution to linear parabolic equation with double degeneracy

We obtain the upper bound for the decay rate of the solution to the Dirichlet initial boundary value problem for a linear parabolic second order equation with a double degeneracy $\mu(x)u_t=(\rho(x)a_{ij}(t,x)u_{x_i})_{x_j}$ in an unbounded domain. For a wide class of revolution domains we prove a lower bound. We adduce the examples showing that the upper and lower bounds are in some sense sharp.
We prove the unique solvability of the problem in an unbounded domain by Galerkin's approximations method.