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Semistrong pulse interactions in a class of coupled reaction-diffusion equations. (English) Zbl 1088.35517
Summary: 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
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