  We study a class of linear parabolic equations in divergence form with
degenerate coefficients on the upper half space. Specifically, the equations
are considered in $(-\infty, T) \times \mathbb{R}^d_+$, where $\mathbb{R}^d_+ =
\{x \in \mathbb{R}^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given, and
the diffusion matrices are the product of $x_d$ and bounded uniformly elliptic
matrices, which are degenerate at $\{x_d=0\}$. As such, our class of equations
resembles well the corresponding class of degenerate viscous Hamilton-Jacobi
equations. We obtain wellposedness results and regularity type estimates in
some appropriate weighted Sobolev spaces for the solutions.
