The problem is from Folland Real Analysis chapter 9 )
Define $G$ on $\mathbb R^n \times\mathbb R $ as $ G(x,t) = (4\pi t)^{\frac{-n}{2}} e^{\frac{-|x|^2}{4t}} \chi_{(0,\infty)}(t) $
a. We are to prove that $ (\partial_t-\Delta)G = \delta $ where $ \Delta $ is the Laplacian on $\mathbb R^n $ and $ \delta $ is the Dirac delta function. (Let $ G^\varepsilon(x,t) = G(x,t)\chi_{(\varepsilon,\infty)}(t) $ then $ G^\varepsilon \to G $ in $ \mathcal{D}' $ (convergence in distributions i.e. continuous linear functionals equipped with the weak topology) , we are to compute $ \langle (\partial_t-\Delta)G^\varepsilon,\varphi\rangle $ for $\varphi \in C_C^\infty $ recalling the discussion of the heat equation)
b. We are to prove that for $\varphi \in C_C^\infty(\mathbb R^n \times\mathbb R) $ the function $ f = G*\varphi $ satisfies $ (\partial_t - \Delta)f = \varphi $
I have also seen Problem on convergence in distributions from Folland's real analysis I understand the concept and I don't have any problem about number b(This part is pretty straight forward).
Here are where my problems are: whenever I am plugging the $G$ function to do in detail, I become stuck. I cannot seem to tackle the part a, as simple as it might sound, I tried doing but always ended up getting some close result but with something wrong. So I really need the help on this in order to do it. I tried asking two people I know and they could not help me either, and of course I appreciate the help on this. Thanks all helpers.