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 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 (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.

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.

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 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 (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 convolution 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.

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 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 (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.