It is known that in the case of more than two independent variables, it is usually not possible (especially in the case of PDE with the variable coefficients) to reduce a linear partial differential equation of the second order to a canonical form throughout a domain. I read the above result, for example, in Renuka Ravindran's book, Partial Differential Equations, Page 58~59. Since I have not found any detailed explanation of this result, I want to construct a counterexample of PDE in at least three independent variables, like this
$$\sum_{i,j=1}^n a_{ij}(x)u_{x_ix_j}+\sum_{i=1}^n b_{i}(x)u_{x_i}+c(x)u(x)=0, x\in\Omega\subset\mathbb{R}^n, $$
where $n\geq3,$
such that which can not be reduced to canonical form globally. But I do not how.  Can anyone help me, or tell me where I can find these counterexamples?