On a class of divergence form linear parabolic equations with degenerate coefficients

Abstract

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.

27 pages