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We consider the following Gierer-Meinhardt system in R:    At = 2A ′′ −A + A p Hq x ∈ (−1, 1), t > 0, τHt = DH ′′ −H + A r Hs x ∈ (−1, 1), t > 0, A ′ (−1) = A(1) = H (−1) = H (1) = 0, where (p, q, r, s) satisfy 1 < qr (s + 1)(p− 1) < +∞, 1 < p < +∞, and where 2 << 1, 0 < D < ∞, τ ≥ 0, D and τ are constants which are independent of 2. We give a rigorous and unified approach to show that the existence and stability of N−peaked steady-states can be reduced to computing two matrices in terms of the coefficients D,N, p, q, r, s. Moreover, it is shown that N−peaked steady-states are generated by exactly two types of peaks, provided their mutual distance is bounded away from zero..

Existence, Classification and Stability Analysis of Multiple-peaked Solutions for the Gierer-Meinhardt System in R<sup>1</sup>