I am trying to solve the PDE $$ku_{xx}(x,t)=u_t(x,t),$$ given that $u(0,t)=u_0$ and $u(x,0)=0$.
Taking the Fourier transform with respect to $x$ gives, $$\frac{\partial}{\partial t}\hat{u}(w,t)+kw\hat{u}(w,t)=0,$$ where $\hat{u}(w,t)=\mathcal{F}_x(u(x,t)).$ The solution to this quasi-ODE is $$\hat{u}(w,t)=A(w)e^{-kw^2t}.$$ But I am unsure how to progress from here.