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

TL;DR

This paper establishes well-posedness and regularity results for a class of degenerate second-order linear parabolic equations in divergence form on the upper half space, with coefficients that degenerate near the boundary.

Contribution

It introduces new well-posedness and regularity results for degenerate parabolic equations with partially VMO coefficients in weighted Sobolev spaces.

Findings

Solutions are well-posed under degenerate conditions.

Regularity results are obtained in weighted Sobolev spaces.

Results extend to systems of equations.

Abstract

We study a class of second-order degenerate linear parabolic equations in divergence form in $(-\infty, T) \times \mathbb R^d_+$ with homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial \mathbb R^d_+$, where $\mathbb R^d_+ = \{x \in \mathbb R^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given. The coefficient matrices of the equations are the product of $\mu(x_d)$ and bounded uniformly elliptic 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. Under a partially VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems.