Consider $ u_t + uu_x =0 $ and an associated problem $uu_t + u^2u_x=0$ ($*$). If we let $ w= u^2$, the second PDE becomes
$$ w_t + \left(\dfrac{2}{3}w^{3/2}\right)_x=0 \>. (**)$$
I am required to show that $u$ is a smooth solution of ($*$) with $u(x,0)=u_0(x)$ iff $w=u^2$ is a smooth solution of ($**$) with $w(x,0)=u^2(x,0)$.
I'm not quite sure where to start and would appreciate a nudge in the correct direction.
Thank you