Problem: Let $v(t,x)$ solve the following initial-value problem $$ex seddm{ p_tv-lap v=0,\ v|_{t=0}=u. } eex$$ Show that $v(t,x)$ has the representation $v(t,x)=e^{tlap}u(x)$. Here, $e^{tlap}u(x)$ is defined through the Fourier transform by $e^{tlap}u=calF^{-1}(e^{-t|xi|^2} calF u)$.
Proof: $$eexea &quad p_that v+|xi|^2hat v=0, hat v|_{t=0}=hat u\ &
a hat v=e^{-t|xi|^2}hat u\ &
a v=calF^{-1}sex{e^{-t|xi|^2}hat u} equiv e^{tlap}uqwz{Fourier multiplier}. eeaeeex$$