Semistrong pulse interactions in a class of coupled reaction-diffusion equations.

*(English)*Zbl 1088.35517Summary: Pulse-pulse interactions play central roles in a variety of pattern formation phenomena, including self-replication. In this article, we develop a theory for the semistrong interaction of pulses in a class of singularly perturbed coupled reaction-diffusion equations that includes the (generalized) Gierer-Meinhardt, Gray-Scott, Schnakenberg, and Thomas models, among others. Geometric conditions are determined on the reaction kinetics for whether the pulses in a two-pulse solution attract or repel, and ODEs are derived for the time-dependent separation distance between their centers and for their wave speeds. In addition, conditions for the existence of stationary two-pulse solutions are identified, and the interactions between stationary and dynamically evolving two-pulse solutions are studied. The theoretical results are illustrated in a series of examples. In two of these, which are related to the classical Gierer-Meinhardt equation, we find that the pulse amplitudes blow up in finite time. Moreover, the blow-up of stationary one-pulse solutions and of dynamically varying two-pulse solutions occurs precisely at the parameter values for which the theory we develop predicts that these solutions should cease to exist as bounded solutions. Finally, generalizations to \(N\)-pulse solutions are presented.

Reviewer: Reviewer (Berlin)

##### MSC:

35K57 | Reaction-diffusion equations |

35K45 | Initial value problems for second-order parabolic systems |

35B25 | Singular perturbations in context of PDEs |

35B32 | Bifurcations in context of PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |

34C30 | Manifolds of solutions of ODE (MSC2000) |

34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |

92E20 | Classical flows, reactions, etc. in chemistry |