Transported step forcing in an advection-diffusion equation ($u(t,x)=-\frac{1}{2}(x-t{)}_{+}^2$)

In coordinates $s=t$, $y=x-t$, the transport derivative $u_t+u_x$ becomes $U_s$. A Heaviside step function moving at that speed becomes stationary, and its twice-integrated profile gives the displayed continuously differentiable solution of $u_t+u_x=u_{xx}+H(x-t)$. Its second spatial derivative has a jump, so the equation holds away from the moving interface and in the distributional sense there.