MHB Solution given by sum of functions on a PDE

[Markov2]

Consider $u_t+u_x=g(x),\,x\in\mathbb R,\,t>0$ and $u(x,0)=f(x).$ Given $f,g\in C^1,$ then show that $u(x,t)$ has the form $u(x,t)=f(x-t)+\sqrt{2\pi}(g*h)(x)$ where $h(x)=\chi_{[0,t]}(x).$

So we just apply the Fourier transform to get $\dfrac{{\partial U}}{{\partial t}} + iwU = U(g)$ and $U((x,0))=U(f ),$ so by solving the ODE I get $U(u)(w,t)=ce^{-iwt}+\dfrac{U(g)(w,t)}{iw},$ now I'm quite confusing on placing the initial condition, what's the correct way?

 

[Markov2]

I'm quite interested on this one, I want to know how to use the initial condition and plug it into $U(u)(w,t),$ please! :D