Upper and lower bounds for heat kernels with very strong degeneracy

We consider the following degenerate parabolic equation on a bounded domain of ${\mathbb{R}}^d$ containing $0$: ${\mathrm{\partial }}_{t}u-\mathrm{\nabla }\cdot (\phi (t,x)\mathrm{\nabla }u)=0$, where $\phi$ is a rough weight behaving like $\mathrm{\mid }x{\mathrm{\mid }}^{\theta }$ 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.