In standard PDE textbook, we can find the following result. For the following linear second order partial differential equation $$ \sum_{i,j=1}^n a_{ij}(x)u_{x_ix_j}+\sum_{i=1}^n b_i(x)u_{x_i}+c(x)u(x)=f(x), x\in\Omega\subset \mathbb{R}^n,$$ it is possible to construct a global $C^2$ transformation to reduce it to canonical form, if the coefficients $a_{ij}, b_i, c$ are only constants.
It is impossible to construct a global transformation to reduce the above PDE, if the coefficients $a_{ij}, b_i, c$ are not all constant functions. But I can not find a counterexample.
Can anyone help me to construct a simple counterexample, or tell me where I can find the counterexample. Thanks a lot.
I don't know which book you are reading, but I think if you know the reason for this result, you don't need counterexamples to help you understand.
Actually, if $u\in C^1$, the given PDE could be transformed to the divergence form. Then you can multiply a $v \in C^{\infty}_0$ to change this equation to its weak form. After that, you may use your bilinear form knowledge to find the coordinate transformation $P$ which could change this PDE to diagonal matrix, and meantime you get $P \nabla u$. If $P$ could pass though $\nabla$, then you can find the canonical form, otherwise, you can't do that.
Obviously, if the coefficients are constant, $P$ would be a scalar matrix which would freely pass through $\nabla$.
I don't know if you understand what I mean. My English might be not good enough to express it.