When a PDE is solved via Fourier transform, is there already a uniqueness assertion that comes for free?
For example, if we Fourier the heat equation \begin{align} \partial_t u(x,t) &= \Delta u(x,t)\\ u(x,0) &= u^0(x) \end{align} to get in Fourier space (assuming $u^0 \in L^1$) \begin{align} \partial_t\hat{u}(\xi,t) &= -|\xi|^2\hat{u}(\xi,t)\\ \hat{u}(\xi,0) &= \hat{u^0}(\xi) \end{align} we know that the Fourier equation has a unique solution $\hat{u}(\xi,t) = e^{-t|\xi|^2} \hat{u^0}(\xi)$.
I suppose my question is: If the Fourier transform of a PDE (not just the heat equation) admits a unique solution, does this mean that when we Fourier back, the solution we get is unique? And, in what sense is the solution unique (in $L^1$?).