  We consider in this paper a diffusion-convection reaction equation in one
space dimension. The main assumptions are about the reaction term, which is
monostable, and the diffusivity, which changes sign once or twice; then, we
deal with a forward-backward parabolic equation. Our main results concern the
existence of globally defined traveling waves, which connect two equilibria and
cross both regions where the diffusivity is positive and regions where it is
negative. We also investigate the monotony of the profiles and show the
appearance of sharp behaviours at the points where the diffusivity degenerates.
In particular, if such points are interior points, then the sharp behaviours
are new and unusual.
