Front blocking and propagation in cylinders with varying cross section

  • Henri BerestyckiEmail author
  • Juliette Bouhours
  • Guillemette Chapuisat


In this paper we consider a bistable reaction–diffusion equation in unbounded domains and we investigate geometric conditions under which propagation, possibly partial, takes place in some direction or, on the contrary, there is a blocking phenomenon. We start by proving the well-posedness of the problem. Then we prove that when the domain has a decreasing cross section with respect to the direction of propagation there is complete propagation. Further, we prove that the wave can be blocked as it comes through an abrupt opening. Finally we discuss various general geometrical properties that ensure either partial or complete invasion by 1. In particular, we show that in a domain that is “star-shaped” with respect to an axis, there is complete invasion by 1.

Mathematics Subject Classification

35B08 35B30 35B40 35C07 35K57 92B05 92C20 



The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 321186—ReaDi—“Reaction–Diffusion Equations, Propagation and Modelling” held by Henri Berestycki. This work was also partially supported by the French National Research Agency (ANR), within the project NONLOCAL ANR-14-CE25-0013. This work was initiated while Henri Berestycki was visiting the Department of Mathematics at the University of Chicago and he was also partly supported by an NSF FRG grant DMS-1065979. Juliette Bouhours was supported by a PhD fellowship “Bourse hors DIM” of the “Région Ile de France”.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Henri Berestycki
    • 1
    Email author
  • Juliette Bouhours
    • 1
    • 2
  • Guillemette Chapuisat
    • 1
    • 3
  1. 1.Ecole des Hautes Etudes en Sciences SocialesPSL Research University, CNRS, Centre d’Analyse et Mathématique SocialesParisFrance
  2. 2.Sorbonne Universités UPMC, Univ Paris 06, CNRS Laboratoire Jacques-Louis LionsParisFrance
  3. 3.Aix-Marseille Université, CNRS, Centrale Marseille, Institut de Mathématique de Marseille, UMR 7373MarseilleFrance

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