I just met this problem from Folland's real analysis involving the theory of distributions (generalized functions) and their Fourier transform, exercise 15 page 291 which reads:
Define G on $ R^n \times 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 $ R^n $ and $ \delta $ is the Dirac delta function. (Let $ G^{\epsilon}(x,t) = G(x,t)\chi_{(\epsilon,\infty)}(t) $ then $ G^{\epsilon}\to G $ in $ \mathcal{D}' $ (convergence in distributions i.e. continuous linear functionals equipped with the weak topology) , we are to compute $ <(\partial_t-\Delta)G^{\epsilon},\phi> $ for $\phi \in C_{C}^{\infty} $ recalling the discussion of the heat equation)
b. We are to prove that for $\phi \in C_{C}^{\infty}(R^n \times R) $ the function $ f = G*\phi $ satisfies $ (\partial_t - \Delta)f = \phi $
To be honest I cannot find a way of thinking about either part or of connecting them together so I am in need of help doing both of them. Thanks all.
$\def\R{\mathbf R}\def\O{{\R^n \times \R}}$Let $\phi \in C_c^\infty(\O)$, then \begin{align*}\def\eps{\varepsilon} \def\<#1>{\left<#1\right>}\<\phi, G^\eps> &= \int_\O G^\eps(x,t)\phi(x,t)\, d(x,t)\\ &= \int_\eps^\infty \int_{\R^n} G(x,t)\phi(x,t)\, d(x,t)\\ &\to \int_0^\infty \int_{\R^n} G(x,t)\phi(x,t)\, d(x,t)\\ &= \<\phi, G> \end{align*} Therefore $G^\eps \to G$ in $\mathcal D'(\O)$. We have - just compute - $\Delta G^\eps(x,t) = \partial_t G^\eps(x,t)$ for $t > \eps$. Hence \begin{align*} \<\phi, (\partial_t - \Delta)G^\eps> &= -\<(\partial_t + \Delta)\phi, G^\eps>\\ &= -\int_\O \partial_t \phi G^\eps + \Delta\phi G^\eps\, d(x,t)\\ &= -\int_{t > \eps} \partial_t \phi G^\eps + \Delta\phi G^\eps\, d(x,t)\\ &= \int_{t > \eps} \phi \partial_t G^\eps - \phi\Delta G^\eps\, d(x,t) + \int_{\mathbf R^n} \phi(x,\eps)G(x,\eps)\, dx\\ &= \int_{\mathbf R^n} \phi(x,\eps)G(x,\eps)\, dx\\ &\to \phi(x,0) = \<\phi, \delta>, \qquad \eps \to 0 \end{align*} (the latter is known from the discussion of the heat equation, I suppose).
(2) is the usual nonsense, we have that $$ (\partial_t - \Delta)(G *\phi) = \bigl((\partial_t - \Delta)G\bigr)*\phi = \delta * \phi = \phi $$