Even if this question has an [accepted answer](https://mathoverflow.net/a/446471/113756) from Iosif, I'd like to point out that the conditions on the boundary of $\Omega$ can be *considerably relaxed* for the standard heat equation. Precisely for the mixed problem above, in reference [1] [Hidehiko Yamabe](https://en.wikipedia.org/wiki/Hidehiko_Yamabe) is able to define *on every open bounded open set $\Omega$ in* $\Bbb R^n$ a "kernel function" $\mathscr{K}(x,y,t)$  which coincides with the ordinary Green function on [regular boundary points](https://encyclopediaofmath.org/wiki/Regular_boundary_point) (in Wiener's sense), i.e.
* $\mathscr{K}(x,y,t)$ is a solution of the homogeneous heat equation
* $\mathscr{K}(x,y,t)\to 0$ if $x$ or $y$ tend to a regular boundary,
* if $\varphi\in C(\overline{\Omega})$ then
$$
\lim_{t\to0}\int\limits_D \mathscr{K}(x,y,t)\varphi(y)\mathrm{d}y=\varphi(x)
$$

The construction of $\mathscr{K}(x,y,t)$ is explicit in the sense that it is the limit as $m$ tends to infinity of the composition products of $m$ standard heat kernels. Finally, I can also point out that in the second part of the paper (published later) Yamabe uses $\mathscr{K}(x,y,t)$ to construct a generalized Green's kernel for the Dirichlet problem for Laplace's equation. Well, my two cents.

**References**

[1] Hidehiko Yamabe, "[Kernel functions of diffusion equations. I.](https://projecteuclid.org/journals/osaka-mathematical-journal/volume-9/issue-2/Kernel-functions-of-diffusion-equations-I/ojm/1200689163.full?tab=ArticleLink)" (English) 
Osaka Math. J. 9, 201-214 (1957), [MR0104051](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0104051), [Zbl 0081.31302](https://zbmath.org/0081.31302).

**Note added after a re-reading of the paper**

After a re-reading of the paper I noticed that the description of the method used by Yamabe in the construction of the kernel $\mathscr K(x,y,t)$ I gave above is not correct: he does not use the convolution of heat kernels but their *composition product* (in the sense of Volterra). Thus the construction, despite being fairly explicit, is impractical in most cases: possibly this is one of the reasons why this interesting paper is not well known.