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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References
Agmon S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci 2, 151–218 (1975)
Aronson D.G.: Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)
Berestycki, H., Hamel, F.: Generalized travelling waves for reaction–diffusion equations. In: Perspectives in Nonlinear Partial Differential Equations. In Honor of H. Brezis, Contemp. Math., Vol. 446. American Mathematical Society, Providence, 2007
Berestycki, H., Hamel,F.: Generalized Transition Waves and Their Properties, 2010. arxiv.org/abs/1012.0794, preprint
Berestycki H., Hamel F., Matano H.: Bistable traveling waves around an obstacle. Commun. Pure Appl. Math. 62, 729–788 (2009)
Fife P.C., McLeod J.B.: The approach of solutions of non-linear diffusion equations to traveling front solutions. Arch. Ration. Mech. Anal. 65, 335–361 (1977)
Fisher R.: The wave of advance of advantageous genes. Ann. Eugenics 7, 355–369 (1937)
Hamel F.: Nadirashvili, N.: Entire solutions of the KPP equation. Commun. Pure Appl. Math. 52, 1255–1276 (1999)
Hamel F., Nadirashvili N.: Travelling fronts and entire solutions of the Fisher-KPP equation in \({\mathbb{R}^N}\) . Arch. Ration. Mech. Anal. 157, 91–163 (2001)
Húska J., Poláčik P.: Exponential separation and principal Floquet bundles for linear parabolic equations on \({\mathbb{R}^N}\) . Discrete Contin. Dyn. Syst. 20, 81–113 (2008)
Levitan B.M.: Inverse Sturm-Liouville problems. BNU Science press, Utrecht (1987)
Kolmogorov A.N., Petrovskii I.G., Piskunov N.S.: Étude de l’équation de la chaleur avec croissance de la quantit de matière et son application à un problème biologique. Bull. Moskov. Gos. Univ. Mat. Mekh. 1, 1–25 (1937)
Matano, H.: Talks presented at various conferences
Mellet A., Roquejoffre J.-M., Sire Y.: Generalized fronts for one-dimensional reaction–diffusion equations. Discrete Contin. Dyn. Syst. 26, 303–312 (2010)
Nadin, Grégoire, Rossi, Luca: Propagation Phenomena for Time Heterogeneous KPP Reaction–Diffusion Equations, 2011. arxiv.org/abs/1104.3686, preprint
Nolen J., Ryzhik L.: Traveling waves in a one-dimensional heterogeneous medium. Ann. Inst. H. Poincar Anal. Non Linaire 26, 1021–1047 (2009)
Pinchover Y.: Large time behavior of the heat kernel. J. Funct. Anal 206, 191–209 (2004)
Shen W.: Traveling waves in diffusive random media. J. Dyn. Diff. Equ. 16(4), 1011–1060 (2004)
Shu Y., Li W.-T., Liu N.-W.: Generalized fronts in reaction–diffusion equations with mono-stable nonlinearity. Nonlinear Anal. 74, 433–440 (2011)
Zlatoš, A.: Generalized Traveling Waves in Disordered Media: Existence, Uniqueness, and Stability, 2009. arxiv.org/abs/0901.2369, preprint
Zlatoš, A.: Transition Fronts in Inhomogeneous Fisher-KPP Reaction–Diffusion Equations, 2011. arxiv.org/abs/1103.3094, preprint
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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