Existence and non-existence of FisherKPP transition fronts

We consider Fisher-KPP-type reaction-diffusion equations with spatially inhomogeneous re-action 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. 1 Introduction and main results Fisher-KPP traveling fronts in homogeneous media Traveling front solutions of the reaction-diffusion equation ut = uxx + f(u) (1.1) are used to model phenomena in a range of applications from biology to social sciences, an