What is the heat kernel of the Laplacian on $U\subset\mathbf{R}^n$?

Question

I know that the heat kernel of $$\Delta\colon C^\infty(\mathbf{R}^n)\to C^\infty(\mathbf{R}^n)$$ is given by $$q(t,x,y)=(4\pi t)^{-n/2} e^{-(x-y)^2/4t},$$ but what about the heat kernel of $$\Delta_U\colon C^\infty(U)\to C^\infty(U)$$ with $U\subset\mathbf{R}^n$? Is there an explicit formula? I've looked into some papers (e.g. this one), but all I found was that the heat kernel of $\Delta_U$ can be approximated by $q$ for small $t$.