I have a question about Neumann heat kernels and its estimates.

Let $D$ be a domain of $\mathbb{R}^d$. We define the Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^{2}(D)$ as follows: \begin{align*} \mathcal{E}(f,g)&=\frac{1}{2}\int_{D}(\nabla f,\nabla g)\,dx,\quad f,g \in H^{1}(D), \end{align*} where $H^{1}(D)$ is the 1-st order Sobolev space with Neumann boundary condition. From general theory of Dirichlet forms, we can find $L^2$-semigroup $\{T_t\}_{t>0}$ associated with $(\mathcal{E},\mathcal{F})$. In the following, we assume $\{T_t\}_{t>0}$ admits an integral kernel $p_{t}(x,y)$.

If $D$ is bounded and the boundary of $D$ is sufficiently smooth, it is known that $p_{t}(x,y)$ has the following Gaussian estimate: \begin{align} (1)\quad p(t,x,y)\le c_{1}t^{-d/2}\exp(-|x-y|^2/c_{2}t),\quad dx-\text{a.e. } (x,y)\in D\times D,\,0<t\le 1. \end{align}

But if $D$ is unbounded, even if the boundary $\partial D$ is smooth, $p_{t}(x,y)$ does not necessarily have Gaussian estimate like (1).

**My question**

We assume $D$ satisfies the following assumption: there exist closed subsets $\{K_n\}_{n=1}^{\infty}$ of $\bar{D}$ such that

- $K_{1} \subset K_2 \subset \cdots$.
- each $K_{n} \cap \bar{D}$ is non empty bounded and its boundary is smooth.
- $\bar{D}=\bigcup_{n=1}^{\infty}K_n$.

Then, can we show the following assertion?:

For each $n$,there exist some constants $c_{1,K_n}$, $c_{2,K_n}$ such that

\begin{align} (2)\quad p(t,x,y)\le c_{1,K_n}t^{-d/2}\exp(-|x-y|^2/c_{2,K_n}t),\quad dx-\text{a.e. } (x,y)\in K_n\times K_n,\,0<t \le 1. \end{align}

Roughly speaking, (2) is a kind of localized estimate of $p_t(x,y)$.

If you know papers related to (2), please let me know.

upperbound (1) also holds in unbounded domains, it is thelowerbound which is problematic. Are you interested in a two-sided estimate, or just the upper bound? $\endgroup$