Even if this question has an accepted answer 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 is able to define on every 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 (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." (English) Osaka Math. J. 9, 201-214 (1957), MR0104051, Zbl 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.