Upper and lower bounds for heat kernels with very strong degeneracy

Abstract

We consider the following degenerate parabolic equation on a bounded domain of ${\mathbb{R}}^d$ containing $0$: ${\partial }_{t}u-\nabla \cdot (\phi (t,x)\nabla u)=0$, where $\phi$ is a rough weight behaving like $|x{|}^{\theta }$ near $0$ for some $\theta \ge 2$, and $u$ satisfies Neumann boundary condition. We prove that this equation admits a heat kernel with upper and lower Gaussian-type bounds.