In PDE's book from Evans, is said that $$ u(x,t) = \int_0^t \int_{\mathbb{R}^N} \frac{1}{(4\pi (t-s))^{N/2}} e^{-\frac{|x-y|^2}{4(t-s)}} f(y,s) dy ds $$ is a solution of $$ \begin{cases} u_t - \Delta u = f, \mathbb{R}^N \times (0, \infty) \\ u = 0, \mathbb{R}^N \times \{t = 0\} \end{cases} . $$ I would like to know if is it possible to find a explicitly solution for the equation
$$ \begin{cases} u_t - \Delta u = f, \Omega \times (0, \infty) \\ u(x,0) = 0 , \Omega \\ u = 0, \partial \Omega \times \{t > 0\} \\ \end{cases} , $$ where $\Omega \subset \mathbb{R}^N$ is a bounded domain, for example, on a ball. Any refence in this topic is welcome.