Simplify/rewrite integral related to the Gauss kernel

First write \begin{align} -\frac{y\cdot x}{2\alpha} - \frac{|y|^2}{4\alpha}t - \frac{|y|^2}{2}&=-\frac{2\,y\cdot x+|y|^2(t+2\alpha)}{4\alpha}=-\frac{\left|\frac{x}{\sqrt{t+2\alpha}}+y\sqrt{t+2\alpha}\right|^2-\left|\frac{x}{\sqrt{t+2\alpha}}\right|^2}{4\alpha}\,. \end{align} The $dy$-integral then becomes after a variable transformation \begin{align} \exp\Big(\frac{|x|^2}{4\alpha(t+2\alpha)}\Big)\frac{1}{(t+2\alpha)^{n/2}}\int_{\mathbb R^n}\exp\Big(-\frac{\left|\frac{x}{\sqrt{t+2\alpha}}+u\right|^2}{4\alpha}\Big)\,du\,. \end{align} Since we are integrating over the entire $\mathbb R^n$ we can drop the constant $\frac{x}{\sqrt{t+2\alpha}}$ so that this integral becomes \begin{align} \exp\Big(\frac{|x|^2}{4\alpha(t+2\alpha)}\Big)\frac{1}{(t+2\alpha)^{n/2}}\frac{1}{(4\pi\alpha)^{n/2}}\,. \end{align} This should answer question 1. It is known that the $dz$-integral satisfies the heat equation with initial datum $f\,.$ At the moment I don't see a simple PDE that is solved by $\Phi(x,t)\,.$ Where does that problem come from?