It is well known (cf. Evans §2.3.1) that
$$\Phi(x,t):= \frac{1}{(4\pi t)^{n/2}} e^{-\frac{|x|^2}{4t}}$$
is the fundamental solution of the heat equation on $\mathbb{R}^n$ for $t>0$, i.e.,
$$\begin{aligned} \Phi_t - \Delta \Phi &= 0 & & \text{in } \mathbb{R}^n \times (0,\infty), \\ \Phi &= \delta_0 & &\text{on } \mathbb{R}^n \times \{t=0\}. \end{aligned}$$
I often see that the fundamental solution is used on a bounded, open domain $\Omega \subset \mathbb{R}^n$ with smooth boundary. Sometimes it is called the fundamental solution to the heat equation on $\Omega$ with homogeneous Dirichlet boundary condition $\Gamma(x,y,t)$, i.e.,
$$\begin{aligned} \Gamma_t - \Delta \Gamma &= 0 & & \text{in } \Omega \times (0,\infty), \\ \Gamma &= 0 & &\text{on } \partial \Omega \times (0,\infty), \\ \Gamma &= \delta_0 & &\text{on } \Omega \times \{t=0\}. \end{aligned}$$
Other times this function is called the Green function of the heat equation with Dirichlet boundary condition. But usually is not explained where this function comes from. In particular, I'm interested in the existence and regularity of such a fundamental solution. Could someone explain how one can derive these fundamental solutions or does someone know a good reference on this topic?