Heat equation for a 3d vector

Question

The heat equation is equal to: $$ \partial T/ \partial t - \alpha^2\nabla^2 T = 0 $$ for scalar $T(\mathbf x,t)$, the general solution to this differential equation can easily be found by taking a Fourier transform on T. $$ T(\mathbf x,t) \to T(\mathbf\omega,t) $$ Now what would it be if $T$ was a 3D vector aka: $T=T(x,y,z)$ What's the general solution to this differential equation: $$ \partial \mathbf T/ \partial t - \alpha^2\nabla^2 \mathbf T = 0 $$