solve $u_t + x (u_x)^2 = f(x), \ x \in (1,2) ,\ t \in (0,T)$ with $u(x,T) = g(x)$

Question

Assume $f,g$ smooth enough. Is there a chance that the problem below has a unique solution? How to proceed to the analysis of it? $$ u_t + x (u_x)^2 = f(x), \: (x,t) \in (1,2) \times (0,T) $$ with $u(x,T) = g(x)$. thanks.