If $\Omega \subset \mathbb{R}^n$ is a limited domain with a $C^1$ boundary, and $U_T = \{ (x,t) : x \in \Omega, 0 < t< T \}$, I have to show that
$\lim_{\tau \to t^-}\int_\Omega \Phi(x-\xi, t-\tau)u(\xi,\tau)d\xi = u(x,t)$
for all $u \in C(\bar{U}_T)$, where $\Phi$ is the fundamental solution of the heat equation:
$\Phi(x,t) = \dfrac{1}{(4\pi t)^{n/2}} \exp\left({\dfrac{-\|x\|^2}{4t}}\right)$ if $t>0$ and $\Phi(x,t) = 0$ if $t \leq 0$
This is the last question of a problem with 2 questions and the other is only to prove that:
$\Phi_t(x-\xi, t-\tau) - \Delta_x\Phi(x-\xi, t-\tau) = 0$
and
$\Phi_\tau(x-\xi, t-\tau) + \Delta_\xi\Phi(x-\xi, t-\tau) = 0$
but I don't see if those equalities help or not. I tried everything. Someone help?