Is it true that the PDE $x^2u_{t}+uu_{x}=0$ is non-linear and homogeneous?

Question

I just want to check if this is true, my rationale being that, by calling the LHS $F$, the $uu_{x}$ term ensures that $\lambda F(t,x,u,u_t,u_x)\neq F(t,x,\lambda u,\lambda u_t,\lambda u_x)$ (so non-linear), and since the zero function solves the PDE then is homogeneous?