Hopf Bifurcation of Spike Solutions for the Shadow Gierer-Meinhardt Model

this paper is to study oscillatory-type instabilities that occur for spike-type solutions of the limiting form of the GM model obtained by letting D in (1.1 b). The resulting well-known system (cf. [24], [11], [3]), called the shadow GM model, is given by a + a /h #na = 0 , x #h t = -h dx . (1.3 c) In (1.3 c), denotes the volume of ## In the limit # 0, the existence of equilibrium spike-type solutions for (1.3) is now very well-understood. There can be boundary spikes whose support are on (cf. [19]), and interior spikes located strictly inside ## The determination of the equilibrium spike locations for interior spike solutions to (1.3) has been found to be related to certain geometric ball-packing problems (see [2], [9], [13], [22], [25], and the references therein). For the case # = 0, the dynamics and stability of interior spikes is also well-understood. For # = 0, and for certain ranges of the exponents m and p, and the dimension N , an interior one-spike equilibrium solution to (1.3) is unstable, but has an exponentially small growth rate as # 0. This weak instability, known as metastability, results in an asymptotically exponentially slow drift of the spike towards the closest point on the boundary of# (cf. [11], [11], [24]). When # = 0, the motion of a one-spike solution to (1.3) along the boundary of# has also been characterized (cf. [12]). When # = 0, equilibrium solutions for (1.3) with two or more interior spikes are always unstable with an O(1) growth rate as #