  We consider Fisher-KPP-type reaction-diffusion equations with spatially
inhomogeneous reaction rates. We show that a sufficiently strong localized
inhomogeneity may prevent existence of transition-front-type global in time
solutions while creating a global in time bump-like solution. This is the first
example of a medium in which no reaction-diffusion transition front exists. A
weaker localized inhomogeneity leads to existence of transition fronts but only
in a finite range of speeds. These results are in contrast with both Fisher-KPP
reactions in homogeneous media as well as ignition-type reactions in
inhomogeneous media.
