let $\mathbb{H}$ be the hyperbolic plane and let $k(t,x,y)$ be the associated heat kernel.

I am wondering, if for any fixed $y\in M$ and $\epsilon >0$ the function $u_t(x):=k(t,x,y)$ is continuous in $S:=\lbrace x\in M : d(x,y)>\epsilon \rbrace$ uniformly in $t\in(0,\infty)$?

For instance in Euclidean space $\mathbb{R}^n$ the heat kernel is given by $k(t,x,y)=\frac{1}{(4\pi t)^{n/2}}e^{-\frac{\Vert x-y \Vert^2}{4t}}$. Hence for $\mathbb{R}^n$ the claim is true because the function $u_t(x)$ has bounded derivative on S, i.e. $\sup_{t>0} \vert \nabla u_t(x) \vert\leq C$.

There is a formula for hyperbolic plane as well, given by

$$k(t,x,y)=\frac{\sqrt{2}}{(4\pi t)^{3/2}}e^{-\frac{t}{4}}\int\limits_{d(x,y)}^{\infty}\frac{s e^{-\frac{s^2}{4t}}}{\sqrt{\cosh{s}-\cosh{d(x,y)}}}ds$$ but I do not know how to estimate the derivative for this function.

Q: How can I prove/disprove that for any $x^{*}\in\mathbb{H}^2, x^{*}\neq y, \forall \epsilon>0$ $\exists\delta>0,$ s.t. $\vert u_t(x)-u_t(x^{*}) \vert\leq \epsilon$ for all $x\in \mathbb{H^2}$ with $d(x,x^{*})<\delta$? If the above statement is wrong, What happens if we restrict the domain to $t\in (0,T)$ for $T>0$ arbitrary and fixed?

Any help will be very appreciated!

Best wishes