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 $0$: $\partial_t u - \nabla\cdot(φ(t, x) \nabla u)= 0$, where $φ$ is a rough weight behaving like $|x|^θ$ near $0$ for some $θ\geq 2$, and $u$ satisfies Neumann boundary condition. We prove that