Degenerate linear parabolic equations in divergence form on the upper half space

We study a class of second-order degenerate linear parabolic equations in divergence form ( − ∞ , T stretchy="false">) ×<!-- × <mml:msubsup> mathvariant="double-struck">R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on partial-differential mathvariant="normal">∂<!-- ∂ \partial </inline-formula>, where alttext="double-struck d Baseline equals StartSet x element-of colon greater-than 0 EndSet"> = fence="false" stretchy="false">{ x ∈<!-- ∈ <mml:msup> : &gt; 0 stretchy="false">} encoding="application/x-tex">{\mathbb {R}}^d_+ = \{x \in {R}}^d: x_d&gt;0\} and alttext="upper left-parenthesis right-bracket"> stretchy="false">] encoding="application/x-tex">T\in {(-\infty \infty ]} is given. The coefficient matrices the are product alttext="mu right-parenthesis"> μ<!-- μ encoding="application/x-tex">\mu (x_d) bounded uniformly elliptic matrices, behaves like alttext="x alpha"> α<!-- α encoding="application/x-tex">x_d^\alpha for some given alttext="alpha 2 2 encoding="application/x-tex">\alpha (0,2) which alttext="left-brace right-brace"> encoding="application/x-tex">\{x_d=0\} domain. Our main motivation comes from analysis viscous Hamilton-Jacobi equations. Under partially VMO assumption coefficients, we obtain well-posedness regularity solutions weighted Sobolev spaces. results can be readily extended to systems.