Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions

In this paper, the Gierer-Meinhardt model systems with finite diffusion constants in the whole space R-2 is considered. We give a regorous proof on the existence and the stability of a single spike solution, and by using such informations, the repulsive dynamics of the interacting multi single-spike solutions is also shown when distances between spike solutions are sufficiently large. This clarifies some part of the mechanism of the evolutional process of localized patterns appearing in the Gierer-Meinhardt model equations.