I'm trying to solve the following problem:
Assume that we are given $f(t,x) \in C^{\infty}(\mathbb R \times \mathbb R^n)$ such that $f(t, \cdot) \in \mathcal S(\mathbb R^n)$ for all $t>0$ (where $\mathcal S(\mathbb R^n)$ is Schwartz space). Assume furthermore that $f$ is a solution to the heat equation on $\mathbb R^n$ with $ \begin{align} \partial_tf(t,x) &= \Delta f(t,x) \qquad t>0 \\ \lim_{t\to 0} f(t,x) &= g(x) \end{align} $ for some $g\in \mathcal S(\mathbb R^n)$.
I want to show that under these conditions we must have $f(t,x) = (K_t\ast g)(x)$, where $\ast$ denotes convolution and $ K_t(x) = \frac{1}{(4\pi t)^{n/2}}e^{-\langle x, x\rangle/4t}$ is the heat kernel.
What I'm having trouble with is the following: The idea is to consider the Fourier transform $\hat f$ of $f$ and to derive the ordinary differential equation
$\partial_t \hat f(t,k) = - k^2 \hat f(t,k)$
But how do I even know that $\hat f(t,k)$ is differentiable with respect to $t$? What I would like to do is differentiate under the integral sign here:
$\partial_t \hat f(t,k) =\partial_t \int_{\mathbb R^n} f(t,x)e^{-ikx}\, dx$
But I don't know how to justify it. Could anyone help me out?
Thanks a lot! =)