Nonlinear Asymptotic Stability of the Semistrong Pulse Dynamics in a Regularized Gierer-Meinhardt Model

Abstract

(Received December 6, 2005; accepted September 21, 2006; published February 26, 2007)

We use renormalization group (RG) techniques to prove the nonlinear asymptotic stability for the semistrong regime of two-pulse interactions in a regularized Gierer-Meinhardt system. In the semistrong limit the localized activator pulses interact strongly through the slowly varying inhibitor. The interaction is not tail-tail as in the weak interaction limit, and the pulse amplitudes and speeds change as the pulse separation evolves on algebraically slow time scales. In addition the point spectrum of the associated linearized operator evolves with the pulse dynamics. The RG approach employed here validates the interaction laws of quasi-steady two-pulse patterns obtained formally in the literature, and establishes that the pulse dynamics reduce to a closed system of ordinary differential equations for the activator pulse locations. Moreover, we fully justify the reduction to the nonlocal eigenvalue problem (NLEP) showing that the large difference between the quasi-steady NLEP operator and the operator arising from linearization about the pulse is controlled by the resolvent.

Keywords: Gierer-Meinhardt; semistrong interaction; renormalization group; geometric singular perturbation; nonlocal eigenvalue problem (NLEP)