Upper and lower bounds for heat kernels with very strong degeneracy

We consider the following degenerate parabolic equation on a bounded domain of Rᵈ containing $0$: ∂ₜ u - ∇·(φ(t, x) ∇ u)= 0, where $φ$ is a rough weight behaving like |x|^θ near $0$ for some θ≥ 2, and u satisfies Neumann boundary condition. We prove that this equation admits a heat kernel with upper and lower Gaussian-type bounds.