Fourier transform and weak solution

Question

I have the following problem $$u_t-\Delta u=f,$$ for $(x,t)\in \mathbb{R}^n\times (0,\infty)$ and $$u(x,0)=u_0(x)$$ for $x\in\mathbb{R}^n$. The function $f\in L^2(0,T;L^2(\mathbb{R}^n))$.I have to prove that there exist an unique weak solution to the problem via Fourier transform. I've done the following. Consider $F$ as the fourier transform of the function $f$ and denote $U(w)$ the fourier transform of $u(x,t)$, given by $$U(w,t)=\int_{\mathbb{R}^n}e^{-ix\cdot w}u dx,$$ so the equation transforms into $$U_t+|w|^2U=F.$$Now I'm stucked and dont know how to proceed. Any help will be very appreciated!

Answer 1

Its a linear inhomogenous equation in U(t,w). The general solution is a solution of the homgenous equation $$U_h(t) = c e^{- |w|^2 t}$$ plus any special solution eg. $$U_s(t,w) =F(w)/|w|^2 $$