An elementary proof for one-dimensionality of travelling waves in cylinders (Q1969252)

The author studies the problem \[ -\Delta u=c(x)u_{x_1}= f(x,u) \quad\text{in}\quad \Omega= \omega\times \mathbb{R},\;x_1\in \omega, \] \[ {\partial u\over \partial n}=0\text{ on }\partial \omega\times \mathbb{R},\quad u(\pm\infty) =u_\pm. \] \(u\) can be interpreted as a travelling curve of the associated parabolic problem. The question studied is: suppose that \(c\) and \(f\) are independent of \(x^1\) and \(u_{x^1}>0\). Is then \(u\) also dependent of \(x^1\)?