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

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.

This paper supersedes arXiv:2106.07637 [math.AP]; 30 pages