Consider the initial value problem value problem of finding such that
for and ,
for ,
given data .
Use the Fourier transform to derive a formula for a solution to be,
where H_t is the heat kernel and is defined as
So (skipping to the essentials...),
Taking Fourier transforms...
and
Hence our ODE becomes
So it's a bit easier now but I fail to see where the whole convolution part comes in.