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So I apply Fourier transform, then $\mathcal F(u_t)=\mathcal F(u_xx)$ then $\dfrac{{\partial \mathcal F(u)(w,t)}}{{\partial t}} = - {w^2}\mathcal F(u)(w,t)$ so $\mathcal F(u)(w,t)=ce^{-w^2t}$ then $\mathcal F(u)(w,0)=c=xe^{-|x|},$ is the initial condition well put? I'm not sure really, I'm confused.