  We study a class of nondivergence form second-order degenerate linear
parabolic equations in $(-\infty, T) \times {\mathbb R}^d_+$ with the
homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial
{\mathbb R}^d_+$, where ${\mathbb R}^d_+ = \{x =(x_1,x_2,\ldots, x_d) \in
{\mathbb R}^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given. The
coefficient matrices of the equations are the product of $\mu(x_d)$ and bounded
positive definite matrices, where $\mu(x_d)$ behaves like $x_d^\alpha$ for some
given $\alpha \in (0,2)$, which are degenerate on the boundary $\{x_d=0\}$ of
the domain. The divergence form equations in this setting were studied in [14].
Under a partially weighted VMO assumption on the coefficients, we obtain the
wellposedness and regularity of solutions in weighted Sobolev spaces. Our
research program is motivated by the regularity theory of solutions to
degenerate viscous Hamilton-Jacobi equations.
