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

TL;DR

This paper investigates a class of divergence form linear parabolic equations with degenerate coefficients on the upper half space, establishing well-posedness and regularity results in weighted Sobolev spaces.

Contribution

It introduces a novel analysis framework for degenerate parabolic equations with coefficients vanishing at the boundary, providing new well-posedness and regularity results.

Findings

Established well-posedness in weighted Sobolev spaces

Derived regularity estimates for solutions

Analyzed equations resembling degenerate viscous Hamilton-Jacobi equations

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.