JournalsaihpcOnline First11 March 2022

Reaction–diffusion equations in the half-space

  • Henri Berestycki

    Centre d'Analyse et de Mathématique Sociales, Paris, France, Hong Kong University of Science and Technology, China
  • Cole Graham

    Brown University, Providence, United States of America
Reaction–diffusion equations in the half-space cover

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Abstract

We study reaction–diffusion equations of various types in the half-space. For bistable reactions with Dirichlet boundary conditions, we prove conditional uniqueness: there is a unique nonzero bounded steady state which exceeds the bistable threshold on large balls. Moreover, solutions starting from sufficiently large initial data converge to this steady state as tt \to \infty. For compactly supported initial data, the asymptotic speed of this propagation agrees with the unique speed cc_* of the one-dimensional traveling wave. We furthermore construct a traveling wave in the half-plane of speed cc_*.

In parallel, we show analogous results for ignition reactions under both Dirichlet and Robin boundary conditions. Using our ignition construction, we obtain stronger results for monostable reactions with the same boundary conditions. For such reactions, we show in general that there is a unique nonzero bounded steady state. Furthermore, monostable reactions exhibit the hair-trigger effect: every solution with nontrivial initial data converges to this steady state as tt \to \infty. Given compactly supported initial data, this disturbance propagates at a speed cc_* equal to the minimal speed of one-dimensional traveling waves. We also construct monostable traveling waves in the Dirichlet or Robin half-plane with any speed ccc \geq c_*.

Cite this article

Henri Berestycki, Cole Graham, Reaction–diffusion equations in the half-space. Ann. Inst. H. Poincaré Anal. Non Linéaire (2022), published online first

DOI 10.4171/AIHPC/27