Abstract
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 the 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.
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Communicated by P. Rabinowitz
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Nolen, J., Roquejoffre, JM., Ryzhik, L. et al. Existence and Non-Existence of Fisher-KPP Transition Fronts. Arch Rational Mech Anal 203, 217–246 (2012). https://doi.org/10.1007/s00205-011-0449-4
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DOI: https://doi.org/10.1007/s00205-011-0449-4