Abstract
We consider the following Gierer-Meinhardt system in R: $$\left\{\begin{array}{l} {\epsilon ^2 A'' - A + \frac{{A^2 }} {H} = 0,\quad x \in ( - 1,1),} {DH'' - H + A^2 = 0,\quad x \in ( - 1,1),} {A'( - 1) = A'(1) = H'( - 1) = H'(1) = 0,} \end{array}\right. $$ where e > 0 is a small parameter and D > 0 is a constant independent of e. A cluster is a combination of several spikes concentrating at the same point. In this paper, we rigorously show the existence of symmetric and asymmetric multiple clusters. This result is new for systems and seems not to occur for single equations. We reduce the problem to the computation of two matrices which depend on the coefficient D as well as the number of different clusters and the number of spikes within each cluster.
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More From: Nonlinear Differential Equations and Applications NoDEA
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